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Series (mathematics)

From Wikipedia, the free encyclopedia
(Redirected fromInfinite series)
Infinite sum
This article is about infinite sums. For finite sums, seeSummation.
Part of a series of articles about
Calculus
∫abf′(t)dt=f(b)−f(a){\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}

Inmathematics, aseries is, roughly speaking, anaddition ofinfinitely manyterms, one after the other.[1] The study of series is a major part ofcalculus and its generalization,mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures incombinatorics throughgenerating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such asphysics,computer science,statistics andfinance.

Among theAncient Greeks, the idea that apotentially infinitesummation could produce a finite result was consideredparadoxical, most famously inZeno's paradoxes.[2][3] Nonetheless, infinite series were applied practically by Ancient Greek mathematicians includingArchimedes, for instance in thequadrature of the parabola.[4][5] The mathematical side of Zeno's paradoxes was resolved using the concept of alimit during the 17th century, especially through the early calculus ofIsaac Newton.[6] The resolution was made more rigorous and further improved in the 19th century through the work ofCarl Friedrich Gauss andAugustin-Louis Cauchy,[7] among others, answering questions about which of these sums exist via thecompleteness of the real numbers and whether series terms can be rearranged or not without changing their sums usingabsolute convergence andconditional convergence of series.

In modern terminology, any orderedinfinite sequence(a1,a2,a3,…){\displaystyle (a_{1},a_{2},a_{3},\ldots )} of terms, whether those terms are numbers,functions,matrices, or anything else that can be added, defines a series, which is the addition of the⁠ai{\displaystyle a_{i}}⁠ one after the other. To emphasize that there are an infinite number of terms, series are often also calledinfinite series to contrast withfinite series, a term sometimes used forfinite sums. Series are represented by anexpression likea1+a2+a3+⋯,{\displaystyle a_{1}+a_{2}+a_{3}+\cdots ,}or, usingcapital-sigma summation notation,[8]∑i=1∞ai.{\displaystyle \sum _{i=1}^{\infty }a_{i}.}

The infinite sequence of additions expressed by a series cannot be explicitly performed in sequence in a finite amount of time. However, if the terms and their finite sums belong to aset that haslimits, it may be possible to assign a value to a series, called thesum of the series. This value is the limit as⁠n{\displaystyle n}⁠ tends toinfinity of the finite sums of the⁠n{\displaystyle n}⁠ first terms of the series if the limit exists.[9][10][11] These finite sums are called thepartial sums of the series. Using summation notation,∑i=1∞ai=limn→∞∑i=1nai,{\displaystyle \sum _{i=1}^{\infty }a_{i}=\lim _{n\to \infty }\,\sum _{i=1}^{n}a_{i},}if it exists.[9][10][11] When the limit exists, the series isconvergent orsummable and also the sequence(a1,a2,a3,…){\displaystyle (a_{1},a_{2},a_{3},\ldots )} issummable, and otherwise, when the limit does not exist, the series isdivergent.[9][10][11]

The expression∑i=1∞ai{\textstyle \sum _{i=1}^{\infty }a_{i}} denotes both the series—the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the explicit limit of the process. This is a generalization of the similar convention of denoting bya+b{\displaystyle a+b} both theaddition—the process of adding—and its result—thesum of⁠a{\displaystyle a}⁠ and⁠b{\displaystyle b}⁠.

Commonly, the terms of a series come from aring, often thefieldR{\displaystyle \mathbb {R} } of thereal numbers or the fieldC{\displaystyle \mathbb {C} } of thecomplex numbers. If so, the set of all series is also itself a ring, one in which the addition consists of adding series terms together term by term and the multiplication is theCauchy product.[12][13][14]

Definition

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Series

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Aseries or, redundantly, aninfinite series, is an infinite sum. It is often represented as[8][15][16]a0+a1+a2+⋯ora1+a2+a3+⋯,{\displaystyle a_{0}+a_{1}+a_{2}+\cdots \quad {\text{or}}\quad a_{1}+a_{2}+a_{3}+\cdots ,}where thetermsak{\displaystyle a_{k}} are the members of asequence ofnumbers,functions, or anything else that can beadded. A series may also be represented withcapital-sigma notation:[8][16]∑k=0∞akor∑k=1∞ak.{\displaystyle \sum _{k=0}^{\infty }a_{k}\qquad {\text{or}}\qquad \sum _{k=1}^{\infty }a_{k}.}

It is also common to express series using a few first terms, an ellipsis, a general term, and then a final ellipsis, the general term being an expression of the⁠n{\displaystyle n}⁠th term as afunction of⁠n{\displaystyle n}⁠:a0+a1+a2+⋯+an+⋯ or f(0)+f(1)+f(2)+⋯+f(n)+⋯.{\displaystyle a_{0}+a_{1}+a_{2}+\cdots +a_{n}+\cdots \quad {\text{ or }}\quad f(0)+f(1)+f(2)+\cdots +f(n)+\cdots .}For example,Euler's number can be defined with the series∑n=0∞1n!=1+1+12+16+⋯+1n!+⋯,{\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n!}}=1+1+{\frac {1}{2}}+{\frac {1}{6}}+\cdots +{\frac {1}{n!}}+\cdots ,} wheren!{\displaystyle n!} denotes the product of then{\displaystyle n} firstpositive integers, and0!{\displaystyle 0!} is conventionally equal to1.{\displaystyle 1.}[17][18][19]

Partial sum of a series

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Given a seriess=∑k=0∞ak{\textstyle s=\sum _{k=0}^{\infty }a_{k}}, its⁠n{\displaystyle n}⁠thpartial sum is[9][10][11][16]sn=∑k=0nak=a0+a1+⋯+an.{\displaystyle s_{n}=\sum _{k=0}^{n}a_{k}=a_{0}+a_{1}+\cdots +a_{n}.}

Some authors directly identify a series with its sequence of partial sums.[9][11] Either the sequence of partial sums or the sequence of terms completely characterizes the series, and the sequence of terms can be recovered from the sequence of partial sums by taking the differences between consecutive elements,an=sn−sn−1.{\displaystyle a_{n}=s_{n}-s_{n-1}.}

Partial summation of a sequence is an example of a linearsequence transformation, and it is also known as theprefix sum incomputer science. The inverse transformation for recovering a sequence from its partial sums is thefinite difference, another linear sequence transformation.

Partial sums of series sometimes have simpler closed form expressions, for instance anarithmetic series has partial sumssn=∑k=0n(a+kd)=a+(a+d)+(a+2d)+⋯+(a+nd)=(n+1)(a+12nd),{\displaystyle s_{n}=\sum _{k=0}^{n}\left(a+kd\right)=a+(a+d)+(a+2d)+\cdots +(a+nd)=(n+1){\bigl (}a+{\tfrac {1}{2}}nd{\bigr )},}and ageometric series has partial sums[20][21][22]sn=∑k=0nark=a+ar+ar2+⋯+arn=a1−rn+11−r{\displaystyle s_{n}=\sum _{k=0}^{n}ar^{k}=a+ar+ar^{2}+\cdots +ar^{n}=a{\frac {1-r^{n+1}}{1-r}}}if⁠r≠1{\displaystyle r\neq 1}⁠ or simply⁠sn=a(n+1){\displaystyle s_{n}=a(n+1)}⁠ if⁠r=1{\displaystyle r=1}⁠.

Sum of a series

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Illustration of 3geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.

Strictly speaking, a series is said toconverge, to beconvergent, or to besummable when the sequence of its partial sums has alimit. When the limit of the sequence of partial sums does not exist, the seriesdiverges or isdivergent.[23] When the limit of the partial sums exists, it is called thesum of the series orvalue of the series:[9][10][11][16]∑k=0∞ak=limn→∞∑k=0nak=limn→∞sn.{\displaystyle \sum _{k=0}^{\infty }a_{k}=\lim _{n\to \infty }\sum _{k=0}^{n}a_{k}=\lim _{n\to \infty }s_{n}.}A series with only a finite number of nonzero terms is always convergent. Such series are useful for considering finite sums without taking care of the numbers of terms.[24] When the sum exists, the difference between the sum of a series and itsn{\displaystyle n}th partial sum,s−sn=∑k=n+1∞ak,{\textstyle s-s_{n}=\sum _{k=n+1}^{\infty }a_{k},} is known as then{\displaystyle n}thtruncation error of the infinite series.[25][26]

An example of a convergent series is the geometric series1+12+14+18+⋯+12k+⋯.{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{k}}}+\cdots .}

It can be shown by algebraic computation that each partial sumsn{\displaystyle s_{n}} is∑k=0n12k=2−12n.{\displaystyle \sum _{k=0}^{n}{\frac {1}{2^{k}}}=2-{\frac {1}{2^{n}}}.} As one haslimn→∞(2−12n)=2,{\displaystyle \lim _{n\to \infty }\left(2-{\frac {1}{2^{n}}}\right)=2,}the series is convergent and converges to⁠2{\displaystyle 2}⁠ with truncation errors1/2n{\textstyle 1/2^{n}}.[20][21][22]

By contrast, the geometric series∑k=0∞2k{\displaystyle \sum _{k=0}^{\infty }2^{k}}is divergent in thereal numbers.[20][21][22] However, it is convergent in theextended real number line, with+∞{\displaystyle +\infty } as its limit and+∞{\displaystyle +\infty } as its truncation error at every step.[27]

When a series's sequence of partial sums is not easily calculated and evaluated for convergence directly,convergence tests can be used to prove that the series converges or diverges.

Grouping and rearranging terms

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Grouping

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In ordinaryfinite summations, terms of the summation can be grouped and ungrouped freely without changing the result of the summation as a consequence of theassociativity of addition.a0+a1+a2={\displaystyle a_{0}+a_{1}+a_{2}={}}a0+(a1+a2)={\displaystyle a_{0}+(a_{1}+a_{2})={}}(a0+a1)+a2.{\displaystyle (a_{0}+a_{1})+a_{2}.} Similarly, in a series, any finite groupings of terms of the series will not change the limit of the partial sums of the series and thus will not change the sum of the series. However, if an infinite number of groupings is performed in an infinite series, then the partial sums of the grouped series may have a different limit than the original series and different groupings may have different limits from one another; the sum ofa0+a1+a2+⋯{\displaystyle a_{0}+a_{1}+a_{2}+\cdots } may not equal the sum ofa0+(a1+a2)+{\displaystyle a_{0}+(a_{1}+a_{2})+{}}(a3+a4)+⋯.{\displaystyle (a_{3}+a_{4})+\cdots .}

For example,Grandi's series⁠1−1+1−1+⋯{\displaystyle 1-1+1-1+\cdots }⁠ has a sequence of partial sums that alternates back and forth between⁠1{\displaystyle 1}⁠ and⁠0{\displaystyle 0}⁠ and does not converge. Grouping its elements in pairs creates the series(1−1)+(1−1)+(1−1)+⋯={\displaystyle (1-1)+(1-1)+(1-1)+\cdots ={}}0+0+0+⋯,{\displaystyle 0+0+0+\cdots ,} which has partial sums equal to zero at every term and thus sums to zero. Grouping its elements in pairs starting after the first creates the series1+(−1+1)+{\displaystyle 1+(-1+1)+{}}(−1+1)+⋯={\displaystyle (-1+1)+\cdots ={}}1+0+0+⋯,{\displaystyle 1+0+0+\cdots ,} which has partial sums equal to one for every term and thus sums to one, a different result.

In general, grouping the terms of a series creates a new series with a sequence of partial sums that is asubsequence of the partial sums of the original series. This means that if the original series converges, so does the new series after grouping: all infinite subsequences of a convergent sequence also converge to the same limit. However, if the original series diverges, then the grouped series do not necessarily diverge, as in this example of Grandi's series above. However, divergence of a grouped series does imply the original series must be divergent, since it proves there is a subsequence of the partial sums of the original series which is not convergent, which would be impossible if it were convergent. This reasoning was applied inOresme's proof of the divergence of the harmonic series,[28] and it is the basis for the generalCauchy condensation test.[29][30]

Rearrangement

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In ordinary finite summations, terms of the summation can be rearranged freely without changing the result of the summation as a consequence of thecommutativity of addition.a0+a1+a2={\displaystyle a_{0}+a_{1}+a_{2}={}}a0+a2+a1={\displaystyle a_{0}+a_{2}+a_{1}={}}a2+a1+a0.{\displaystyle a_{2}+a_{1}+a_{0}.} Similarly, in a series, any finite rearrangements of terms of a series does not change the limit of the partial sums of the series and thus does not change the sum of the series: for any finite rearrangement, there will be some term after which the rearrangement did not affect any further terms: any effects of rearrangement can be isolated to the finite summation up to that term, and finite summations do not change under rearrangement.

However, as for grouping, an infinitary rearrangement of terms of a series can sometimes lead to a change in the limit of the partial sums of the series. Series with sequences of partial sums that converge to a value but whose terms could be rearranged to a form a series with partial sums that converge to some other value are calledconditionally convergent series. Those that converge to the same value regardless of rearrangement are calledunconditionally convergent series.

For series of real numbers and complex numbers, a seriesa0+a1+a2+⋯{\displaystyle a_{0}+a_{1}+a_{2}+\cdots } is unconditionally convergentif and only if the series summing theabsolute values of its terms,|a0|+|a1|+|a2|+⋯,{\displaystyle |a_{0}|+|a_{1}|+|a_{2}|+\cdots ,} is also convergent, a property calledabsolute convergence. Otherwise, any series of real numbers or complex numbers that converges but does not converge absolutely is conditionally convergent. Any conditionally convergent sum of real numbers can be rearranged to yield any other real number as a limit, or to diverge. These claims are the content of theRiemann series theorem.[31][32][33]

A historically important example of conditional convergence is thealternating harmonic series,

∑n=1∞(−1)n+1n=1−12+13−14+15−⋯,{\displaystyle \sum \limits _{n=1}^{\infty }{(-1)^{n+1} \over n}=1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots ,}which has a sum of thenatural logarithm of 2, while the sum of the absolute values of the terms is theharmonic series,∑n=1∞1n=1+12+13+14+15+⋯,{\displaystyle \sum \limits _{n=1}^{\infty }{1 \over n}=1+{1 \over 2}+{1 \over 3}+{1 \over 4}+{1 \over 5}+\cdots ,}which diverges per the divergence of the harmonic series,[28] so the alternating harmonic series is conditionally convergent. For instance, rearranging the terms of the alternating harmonic series so that each positive term of the original series is followed by two negative terms of the original series rather than just one yields[34]1−12−14+13−16−18+15−110−112+⋯=(1−12)−14+(13−16)−18+(15−110)−112+⋯=12−14+16−18+110−112+⋯=12(1−12+13−14+15−16+⋯),{\displaystyle {\begin{aligned}&1-{\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{3}}-{\frac {1}{6}}-{\frac {1}{8}}+{\frac {1}{5}}-{\frac {1}{10}}-{\frac {1}{12}}+\cdots \\[3mu]&\quad =\left(1-{\frac {1}{2}}\right)-{\frac {1}{4}}+\left({\frac {1}{3}}-{\frac {1}{6}}\right)-{\frac {1}{8}}+\left({\frac {1}{5}}-{\frac {1}{10}}\right)-{\frac {1}{12}}+\cdots \\[3mu]&\quad ={\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{6}}-{\frac {1}{8}}+{\frac {1}{10}}-{\frac {1}{12}}+\cdots \\[3mu]&\quad ={\frac {1}{2}}\left(1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}+\cdots \right),\end{aligned}}}which is12{\displaystyle {\tfrac {1}{2}}} times the original series, so it would have a sum of half of the natural logarithm of 2. By the Riemann series theorem, rearrangements of the alternating harmonic series to yield any other real number are also possible.

Operations

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Series addition

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The addition of two seriesa0+a1+a2+⋯{\textstyle a_{0}+a_{1}+a_{2}+\cdots } andb0+b1+b2+⋯{\textstyle b_{0}+b_{1}+b_{2}+\cdots } is given by the termwise sum[13][35][36][37](a0+b0)+(a1+b1)+(a2+b2)+⋯{\textstyle (a_{0}+b_{0})+(a_{1}+b_{1})+(a_{2}+b_{2})+\cdots \,}, or, in summation notation,∑k=0∞ak+∑k=0∞bk=∑k=0∞ak+bk.{\displaystyle \sum _{k=0}^{\infty }a_{k}+\sum _{k=0}^{\infty }b_{k}=\sum _{k=0}^{\infty }a_{k}+b_{k}.}

Using the symbolssa,n{\displaystyle s_{a,n}} andsb,n{\displaystyle s_{b,n}} for the partial sums of the added series andsa+b,n{\displaystyle s_{a+b,n}} for the partial sums of the resulting series, this definition implies the partial sums of the resulting series followsa+b,n=sa,n+sb,n.{\displaystyle s_{a+b,n}=s_{a,n}+s_{b,n}.} Then the sum of the resulting series, i.e., the limit of the sequence of partial sums of the resulting series, satisfieslimn→∞sa+b,n=limn→∞(sa,n+sb,n)=limn→∞sa,n+limn→∞sb,n,{\displaystyle \lim _{n\rightarrow \infty }s_{a+b,n}=\lim _{n\rightarrow \infty }(s_{a,n}+s_{b,n})=\lim _{n\rightarrow \infty }s_{a,n}+\lim _{n\rightarrow \infty }s_{b,n},}when the limits exist. Therefore, first, the series resulting from addition is summable if the series added were summable, and, second, the sum of the resulting series is the addition of the sums of the added series. The addition of two divergent series may yield a convergent series: for instance, the addition of a divergent series with a series of its terms times−1{\displaystyle -1} will yield a series of all zeros that converges to zero. However, for any two series where one converges and the other diverges, the result of their addition diverges.[35]

For series of real numbers or complex numbers, series addition isassociative,commutative, andinvertible. Therefore series addition gives the sets of convergent series of real numbers or complex numbers the structure of anabelian group and also gives the sets of all series of real numbers or complex numbers (regardless of convergence properties) the structure of an abelian group.

Scalar multiplication

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The product of a seriesa0+a1+a2+⋯{\textstyle a_{0}+a_{1}+a_{2}+\cdots } with a constant numberc{\displaystyle c}, called ascalar in this context, is given by the termwise product[35]ca0+ca1+ca2+⋯{\textstyle ca_{0}+ca_{1}+ca_{2}+\cdots }, or, in summation notation,

c∑k=0∞ak=∑k=0∞cak.{\displaystyle c\sum _{k=0}^{\infty }a_{k}=\sum _{k=0}^{\infty }ca_{k}.}

Using the symbolssa,n{\displaystyle s_{a,n}} for the partial sums of the original series andsca,n{\displaystyle s_{ca,n}} for the partial sums of the series after multiplication byc{\displaystyle c}, this definition implies thatsca,n=csa,n{\displaystyle s_{ca,n}=cs_{a,n}} for alln,{\displaystyle n,} and therefore alsolimn→∞sca,n=climn→∞sa,n,{\textstyle \lim _{n\rightarrow \infty }s_{ca,n}=c\lim _{n\rightarrow \infty }s_{a,n},}when the limits exist. Therefore if a series is summable, any nonzero scalar multiple of the series is also summable and vice versa: if a series is divergent, then any nonzero scalar multiple of it is also divergent.

Scalar multiplication of real numbers and complex numbers is associative, commutative, invertible, and itdistributes over series addition.

In summary, series addition and scalar multiplication gives the set of convergent series and the set of series of real numbers the structure of areal vector space. Similarly, one getscomplex vector spaces for series and convergent series of complex numbers. All these vector spaces are infinite dimensional.

Series multiplication

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The multiplication of two seriesa0+a1+a2+⋯{\displaystyle a_{0}+a_{1}+a_{2}+\cdots } andb0+b1+b2+⋯{\displaystyle b_{0}+b_{1}+b_{2}+\cdots } to generate a third seriesc0+c1+c2+⋯{\displaystyle c_{0}+c_{1}+c_{2}+\cdots }, called the Cauchy product,[12][13][14][36][38] can be written in summation notation(∑k=0∞ak)⋅(∑k=0∞bk)=∑k=0∞ck=∑k=0∞∑j=0kajbk−j,{\displaystyle {\biggl (}\sum _{k=0}^{\infty }a_{k}{\biggr )}\cdot {\biggl (}\sum _{k=0}^{\infty }b_{k}{\biggr )}=\sum _{k=0}^{\infty }c_{k}=\sum _{k=0}^{\infty }\sum _{j=0}^{k}a_{j}b_{k-j},}with eachck=∑j=0kajbk−j={\textstyle c_{k}=\sum _{j=0}^{k}a_{j}b_{k-j}={}\!}a0bk+a1bk−1+⋯+ak−1b1+akb0.{\displaystyle \!a_{0}b_{k}+a_{1}b_{k-1}+\cdots +a_{k-1}b_{1}+a_{k}b_{0}.} Here, the convergence of the partial sums of the seriesc0+c1+c2+⋯{\displaystyle c_{0}+c_{1}+c_{2}+\cdots } is not as simple to establish as for addition. However, if both seriesa0+a1+a2+⋯{\displaystyle a_{0}+a_{1}+a_{2}+\cdots } andb0+b1+b2+⋯{\displaystyle b_{0}+b_{1}+b_{2}+\cdots } areabsolutely convergent series, then the series resulting from multiplying them also converges absolutely with a sum equal to the product of the two sums of the multiplied series,[13][36][39]limn→∞sc,n=(limn→∞sa,n)⋅(limn→∞sb,n).{\displaystyle \lim _{n\rightarrow \infty }s_{c,n}=\left(\,\lim _{n\rightarrow \infty }s_{a,n}\right)\cdot \left(\,\lim _{n\rightarrow \infty }s_{b,n}\right).}

Series multiplication of absolutely convergent series of real numbers and complex numbers is associative, commutative, and distributes over series addition. Together with series addition, series multiplication gives the sets of absolutely convergent series of real numbers or complex numbers the structure of acommutativering, and together with scalar multiplication as well, the structure of acommutative algebra; these operations also give the sets of all series of real numbers or complex numbers the structure of anassociative algebra.

Examples of numerical series

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For other examples, seeList of mathematical series andSums of reciprocals § Infinitely many terms.

Pi

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Main articles:Basel problem andLeibniz formula for π

∑n=1∞1n2=112+122+132+142+⋯=π26{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}}

∑n=1∞(−1)n+1(4)2n−1=41−43+45−47+49−411+413−⋯=π{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}(4)}{2n-1}}={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}-\cdots =\pi }

Natural logarithm of 2

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Main article:Natural logarithm of 2 § Series representations

∑n=1∞(−1)n+1n=ln⁡2{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=\ln 2}

∑n=1∞12nn=ln⁡2{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{2^{n}n}}=\ln 2}

Natural logarithm basee

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Main article:e (mathematical constant)

∑n=0∞(−1)nn!=1−11!+12!−13!+⋯=1e{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}=1-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+\cdots ={\frac {1}{e}}}

∑n=0∞1n!=10!+11!+12!+13!+14!+⋯=e{\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots =e}

Convergence testing

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Main article:Convergence tests

One of the simplest tests for convergence of a series, applicable to all series, is thevanishing condition or⁠n{\displaystyle n}⁠th-term test: Iflimn→∞an≠0{\textstyle \lim _{n\to \infty }a_{n}\neq 0}, then the series diverges; iflimn→∞an=0{\textstyle \lim _{n\to \infty }a_{n}=0}, then the test is inconclusive.[46][47]

Absolute convergence tests

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Main article:Absolute convergence

When every term of a series is a non-negative real number, for instance when the terms are theabsolute values of another series of real numbers or complex numbers, the sequence of partial sums is non-decreasing. Therefore a series with non-negative terms converges if and only if the sequence of partial sums is bounded, and so finding a bound for a series or for the absolute values of its terms is an effective way to prove convergence or absolute convergence of a series.[48][49][47][50]

For example, the series1+14+19+⋯+1n2+⋯{\textstyle 1+{\frac {1}{4}}+{\frac {1}{9}}+\cdots +{\frac {1}{n^{2}}}+\cdots \,}is convergent and absolutely convergent because1n2≤1n−1−1n{\textstyle {\frac {1}{n^{2}}}\leq {\frac {1}{n-1}}-{\frac {1}{n}}} for alln≥2{\displaystyle n\geq 2} and atelescoping sum argument implies that the partial sums of the series of those non-negative bounding terms are themselves bounded above by 2.[43] The exact value of this series is16π2{\textstyle {\frac {1}{6}}\pi ^{2}}; seeBasel problem.

This type of bounding strategy is the basis for general series comparison tests. First is the generaldirect comparison test:[51][52][47] For any series∑an{\textstyle \sum a_{n}}, If∑bn{\textstyle \sum b_{n}} is anabsolutely convergent series such that|an|≤C|bn|{\displaystyle \left\vert a_{n}\right\vert \leq C\left\vert b_{n}\right\vert } for some positive real numberC{\displaystyle C} and for sufficiently largen{\displaystyle n}, then∑an{\textstyle \sum a_{n}} converges absolutely as well. If∑|bn|{\textstyle \sum \left\vert b_{n}\right\vert } diverges, and|an|≥|bn|{\displaystyle \left\vert a_{n}\right\vert \geq \left\vert b_{n}\right\vert } for all sufficiently largen{\displaystyle n}, then∑an{\textstyle \sum a_{n}} also fails to converge absolutely, although it could still be conditionally convergent, for example, if thean{\displaystyle a_{n}} alternate in sign. Second is the generallimit comparison test:[53][54] If∑bn{\textstyle \sum b_{n}} is an absolutely convergent series such that|an+1an|≤|bn+1bn|{\displaystyle \left\vert {\tfrac {a_{n+1}}{a_{n}}}\right\vert \leq \left\vert {\tfrac {b_{n+1}}{b_{n}}}\right\vert } for sufficiently largen{\displaystyle n}, then∑an{\textstyle \sum a_{n}} converges absolutely as well. If∑|bn|{\textstyle \sum \left|b_{n}\right|} diverges, and|an+1an|≥|bn+1bn|{\displaystyle \left\vert {\tfrac {a_{n+1}}{a_{n}}}\right\vert \geq \left\vert {\tfrac {b_{n+1}}{b_{n}}}\right\vert } for all sufficiently largen{\displaystyle n}, then∑an{\textstyle \sum a_{n}} also fails to converge absolutely, though it could still be conditionally convergent if thean{\displaystyle a_{n}} vary in sign.

Using comparisons togeometric series specifically,[20][21] those two general comparison tests imply two further common and generally useful tests for convergence of series with non-negative terms or for absolute convergence of series with general terms. First is theratio test:[55][56][57] if there exists a constantC<1{\displaystyle C<1} such that|an+1an|<C{\displaystyle \left\vert {\tfrac {a_{n+1}}{a_{n}}}\right\vert <C} for all sufficiently large n{\displaystyle n}, then∑an{\textstyle \sum a_{n}} converges absolutely. When the ratio is less than1{\displaystyle 1}, but not less than a constant less than1{\displaystyle 1}, convergence is possible but this test does not establish it. Second is theroot test:[55][58][59] if there exists a constantC<1{\displaystyle C<1} such that|an|1/n≤C{\displaystyle \textstyle \left\vert a_{n}\right\vert ^{1/n}\leq C} for all sufficiently large n{\displaystyle n}, then∑an{\textstyle \sum a_{n}} converges absolutely.

Alternatively, using comparisons to series representations ofintegrals specifically, one derives theintegral test:[60][61] iff(x){\displaystyle f(x)} is a positivemonotone decreasing function defined on theinterval[1,∞){\displaystyle [1,\infty )} then for a series with termsan=f(n){\displaystyle a_{n}=f(n)} for all n{\displaystyle n},∑an{\textstyle \sum a_{n}} converges if and only if theintegral∫1∞f(x)dx{\textstyle \int _{1}^{\infty }f(x)\,dx} is finite. Using comparisons to flattened-out versions of a series leads toCauchy's condensation test:[29][30] if the sequence of termsan{\displaystyle a_{n}} is non-negative and non-increasing, then the two series∑an{\textstyle \sum a_{n}} and∑2ka(2k){\textstyle \sum 2^{k}a_{(2^{k})}} are either both convergent or both divergent.

Conditional convergence tests

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Main article:Conditional convergence

A series of real or complex numbers is said to beconditionally convergent (orsemi-convergent) if it is convergent but not absolutely convergent. Conditional convergence is tested for differently than absolute convergence.

One important example of a test for conditional convergence is thealternating series test orLeibniz test:[62][63][64] A series of the form∑(−1)nan{\textstyle \sum (-1)^{n}a_{n}} with allan>0{\displaystyle a_{n}>0} is calledalternating. Such a series converges if the non-negativesequencean{\displaystyle a_{n}} ismonotone decreasing and converges to 0{\displaystyle 0}. The converse is in general not true. A famous example of an application of this test is thealternating harmonic series∑n=1∞(−1)n+1n=1−12+13−14+15−⋯,{\displaystyle \sum \limits _{n=1}^{\infty }{(-1)^{n+1} \over n}=1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots ,}which is convergent per the alternating series test (and its sum is equal to ln⁡2{\displaystyle \ln 2}), though the series formed by taking the absolute value of each term is the ordinaryharmonic series, which is divergent.[65][66]

The alternating series test can be viewed as a special case of the more generalDirichlet's test:[67][68][69] if(an){\displaystyle (a_{n})} is a sequence of terms of decreasing nonnegative real numbers that converges to zero, and(λn){\displaystyle (\lambda _{n})} is a sequence of terms with bounded partial sums, then the series∑λnan{\textstyle \sum \lambda _{n}a_{n}} converges. Takingλn=(−1)n{\displaystyle \lambda _{n}=(-1)^{n}} recovers the alternating series test.

Abel's test is another important technique for handling semi-convergent series.[67][29] If a series has the form∑an=∑λnbn{\textstyle \sum a_{n}=\sum \lambda _{n}b_{n}} where the partial sums of the series with termsbn{\displaystyle b_{n}},sb,n=b0+⋯+bn{\displaystyle s_{b,n}=b_{0}+\cdots +b_{n}} are bounded,λn{\displaystyle \lambda _{n}} hasbounded variation, andlimλnbn{\displaystyle \lim \lambda _{n}b_{n}} exists: ifsupn|sb,n|<∞,{\textstyle \sup _{n}|s_{b,n}|<\infty ,}∑|λn+1−λn|<∞,{\textstyle \sum \left|\lambda _{n+1}-\lambda _{n}\right|<\infty ,} andλnsb,n{\displaystyle \lambda _{n}s_{b,n}}converges, then the series∑an{\textstyle \sum a_{n}} is convergent.

Other specialized convergence tests for specific types of series include theDini test[70] forFourier series.

Evaluation of truncation errors

[edit]

The evaluation of truncation errors of series is important innumerical analysis (especiallyvalidated numerics andcomputer-assisted proof). It can be used to prove convergence and to analyzerates of convergence.

Alternating series

[edit]
Main article:Alternating series

When conditions of thealternating series test are satisfied byS:=∑m=0∞(−1)mum{\textstyle S:=\sum _{m=0}^{\infty }(-1)^{m}u_{m}}, there is an exact error evaluation.[71] Setsn{\displaystyle s_{n}} to be the partial sumsn:=∑m=0n(−1)mum{\textstyle s_{n}:=\sum _{m=0}^{n}(-1)^{m}u_{m}} of the given alternating seriesS{\displaystyle S}. Then the next inequality holds:|S−sn|≤un+1.{\displaystyle |S-s_{n}|\leq u_{n+1}.}

Hypergeometric series

[edit]
Main article:Hypergeometric series

By using theratio, we can obtain the evaluation of the error term when thehypergeometric series is truncated.[72]

Matrix exponential

[edit]
Main article:Matrix exponential

For thematrix exponential:

exp⁡(X):=∑k=0∞1k!Xk,X∈Cn×n,{\displaystyle \exp(X):=\sum _{k=0}^{\infty }{\frac {1}{k!}}X^{k},\quad X\in \mathbb {C} ^{n\times n},}

the following error evaluation holds (scaling and squaring method):[73][74][75]

Tr,s(X):=(∑j=0r1j!(X/s)j)s,‖exp⁡(X)−Tr,s(X)‖≤‖X‖r+1sr(r+1)!exp⁡(‖X‖).{\displaystyle T_{r,s}(X):={\biggl (}\sum _{j=0}^{r}{\frac {1}{j!}}(X/s)^{j}{\biggr )}^{s},\quad {\bigl \|}\exp(X)-T_{r,s}(X){\bigr \|}\leq {\frac {\|X\|^{r+1}}{s^{r}(r+1)!}}\exp(\|X\|).}

Sums of divergent series

[edit]
Main article:Divergent series

Under many circumstances, it is desirable to assign generalized sums to series which fail to converge in the strict sense that their sequences of partial sums do not converge. Asummation method is any method for assigning sums to divergent series in a way that systematically extends the classical notion of the sum of a series. Summation methods includeCesàro summation,generalized Cesàro⁠(C,α){\displaystyle (C,\alpha )}⁠ summation,Abel summation, andBorel summation, in order of applicability to increasingly divergent series. These methods are all based onsequence transformations of the original series of terms or of its sequence of partial sums.A variety of general results concerning possible summability methods are known. TheSilverman–Toeplitz theorem characterizesmatrix summation methods, which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients. The most general methods for summing a divergent series arenon-constructive and concernBanach limits.

Series of functions

[edit]
Main article:Function series

A series of real- or complex-valued functions

∑n=0∞fn(x){\displaystyle \sum _{n=0}^{\infty }f_{n}(x)}

ispointwise convergent to a limit⁠f(x){\displaystyle f(x)}⁠ on a set⁠E{\displaystyle E}⁠ if the series converges for each⁠x{\displaystyle x}⁠ in⁠E{\displaystyle E}⁠ as a series of real or complex numbers. Equivalently, the partial sums

sN(x)=∑n=0Nfn(x){\displaystyle s_{N}(x)=\sum _{n=0}^{N}f_{n}(x)}

converge to⁠f(x){\displaystyle f(x)}⁠ as⁠N{\displaystyle N}⁠ goes to infinity for each⁠x{\displaystyle x}⁠ in⁠E{\displaystyle E}⁠.

A stronger notion of convergence of a series of functions isuniform convergence. A series converges uniformly in a setE{\displaystyle E} if it converges pointwise to the function⁠f(x){\displaystyle f(x)}⁠ at every point ofE{\displaystyle E} and the supremum of these pointwise errors in approximating the limit by the⁠N{\displaystyle N}⁠th partial sum,

supx∈E|sN(x)−f(x)|{\displaystyle \sup _{x\in E}{\bigl |}s_{N}(x)-f(x){\bigr |}}

converges to zero with increasing⁠N{\displaystyle N}⁠,independently of⁠x{\displaystyle x}⁠.

Uniform convergence is desirable for a series because many properties of the terms of the series are then retained by the limit. For example, if a series of continuous functions converges uniformly, then the limit function is also continuous. Similarly, if the⁠fn{\displaystyle f_{n}}⁠ areintegrable on a closed and bounded interval⁠I{\displaystyle I}⁠ and converge uniformly, then the series is also integrable on⁠I{\displaystyle I}⁠ and can be integrated term by term. Tests for uniform convergence includeWeierstrass' M-test,Abel's uniform convergence test,Dini's test, and theCauchy criterion.

More sophisticated types of convergence of a series of functions can also be defined. Inmeasure theory, for instance, a series of functions convergesalmost everywhere if it converges pointwise except on a set ofmeasure zero. Othermodes of convergence depend on a differentmetric space structure on thespace of functions under consideration. For instance, a series of functionsconverges in mean to a limit function⁠f{\displaystyle f}⁠ on a set⁠E{\displaystyle E}⁠ if

limN→∞∫E|sN(x)−f(x)|2dx=0.{\displaystyle \lim _{N\rightarrow \infty }\int _{E}{\bigl |}s_{N}(x)-f(x){\bigr |}^{2}\,dx=0.}

Power series

[edit]
Main article:Power series

Apower series is a series of the form

∑n=0∞an(x−c)n.{\displaystyle \sum _{n=0}^{\infty }a_{n}(x-c)^{n}.}

TheTaylor series at a point⁠c{\displaystyle c}⁠ of a function is a power series that, in many cases, converges to the function in a neighborhood of⁠c{\displaystyle c}⁠. For example, the series

∑n=0∞xnn!{\displaystyle \sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}

is the Taylor series ofex{\displaystyle e^{x}} at the origin and converges to it for every⁠x{\displaystyle x}⁠.

Unless it converges only at⁠x=c{\displaystyle x=c}⁠, such a series converges on a certain open disc of convergence centered at the point⁠c{\displaystyle c}⁠ in the complex plane, and may also converge at some of the points of the boundary of the disc. The radius of this disc is known as theradius of convergence, and can in principle be determined from the asymptotics of the coefficients⁠an{\displaystyle a_{n}}⁠. The convergence is uniform onclosed andbounded (that is,compact) subsets of the interior of the disc of convergence: to wit, it isuniformly convergent on compact sets.

Historically, mathematicians such asLeonhard Euler operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.

Formal power series

[edit]
Main article:Formal power series

While many uses of power series refer to their sums, it is also possible to treat power series asformal sums, meaning that no addition operations are actually performed, and the symbol "+" is an abstract symbol of conjunction which is not necessarily interpreted as corresponding to addition. In this setting, the sequence of coefficients itself is of interest, rather than the convergence of the series.Formal power series are used incombinatorics to describe and studysequences that are otherwise difficult to handle, for example, using the method ofgenerating functions. TheHilbert–Poincaré series is a formal power series used to studygraded algebras.

Even if the limit of the power series is not considered, if the terms support appropriate structure then it is possible to define operations such asaddition,multiplication,derivative,antiderivative for power series "formally", treating the symbol "+" as if it corresponded to addition. In the most common setting, the terms come from acommutative ring, so that the formal power series can be added term-by-term and multiplied via theCauchy product. In this case the algebra of formal power series is thetotal algebra of themonoid ofnatural numbers over the underlying term ring.[76] If the underlying term ring is adifferential algebra, then the algebra of formal power series is also a differential algebra, with differentiation performed term-by-term.

Laurent series

[edit]
Main article:Laurent series

Laurent series generalize power series by admitting terms into the series with negative as well as positive exponents. A Laurent series is thus any series of the form

∑n=−∞∞anxn.{\displaystyle \sum _{n=-\infty }^{\infty }a_{n}x^{n}.}

If such a series converges, then in general it does so in anannulus rather than a disc, and possibly some boundary points. The series converges uniformly on compact subsets of the interior of the annulus of convergence.

Dirichlet series

[edit]
Main article:Dirichlet series

ADirichlet series is one of the form

∑n=1∞anns,{\displaystyle \sum _{n=1}^{\infty }{a_{n} \over n^{s}},}

where⁠s{\displaystyle s}⁠ is acomplex number. For example, if all⁠an{\displaystyle a_{n}}⁠ are equal to⁠1{\displaystyle 1}⁠, then the sum of the Dirichlet series is theRiemann zeta function

ζ(s)=∑n=1∞1ns.{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.}

Like the zeta function, Dirichlet series in general play an important role inanalytic number theory. Generally a Dirichlet series converges if the real part of⁠s{\displaystyle s}⁠ is greater than a number called the abscissa of convergence. In many cases, a function defined by a Dirichlet series is ananalytic function that can be extended outside the domain of convergence of the series byanalytic continuation. For example, the Dirichlet series for the zeta function converges absolutely when⁠Re⁡(s)>1{\displaystyle \operatorname {Re} (s)>1}⁠, but the zeta function can be extended to aholomorphic function defined onC∖{1}{\displaystyle \mathbb {C} \setminus \{1\}} with a simplepole at ⁠1{\displaystyle 1}⁠.

This series can be directly generalized togeneral Dirichlet series.

Trigonometric series

[edit]
Main article:Trigonometric series

A series of functions in which the terms aretrigonometric functions is called atrigonometric series:

A0+∑n=1∞(Ancos⁡nx+Bnsin⁡nx).{\displaystyle A_{0}+\sum _{n=1}^{\infty }\left(A_{n}\cos nx+B_{n}\sin nx\right).}

The most important example of a trigonometric series is theFourier series of a function.

Asymptotic series

[edit]
Main article:Asymptotic expansion

Asymptotic series, typically calledasymptotic expansions, are infinite series whose terms are functions of a sequence of differentasymptotic orders and whose partial sums are approximations of some other function in anasymptotic limit. In general they do not converge, but they are still useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. They are crucial tools inperturbation theory and in theanalysis of algorithms.

An asymptotic series cannot necessarily be made to produce an answer as exactly as desired away from the asymptotic limit, the way that an ordinary convergent series of functions can. In fact, a typical asymptotic series reaches its best practical approximation away from the asymptotic limit after a finite number of terms; if more terms are included, the series will produce less accurate approximations.

History of the theory of infinite series

[edit]

Development of infinite series

[edit]

Infinite series play an important role in modern analysis ofAncient Greekphilosophy of motion, particularly inZeno's paradoxes.[77] The paradox ofAchilles and the tortoise demonstrates that continuous motion would require anactual infinity of temporal instants, which was arguably anabsurdity: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on.Zeno is said to have argued that therefore Achilles couldnever reach the tortoise, and thus that continuous movement must be an illusion. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the purely mathematical and imaginative side of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise. However, in modern philosophy of motion the physical side of the problem remains open, with both philosophers and physicists doubting, like Zeno, that spatial motions are infinitely divisible: hypothetical reconciliations ofquantum mechanics andgeneral relativity in theories ofquantum gravity often introducequantizations ofspacetime at thePlanck scale.[78][79]

Greek mathematicianArchimedes produced the first known summation of an infinite series with amethod that is still used in the area of calculus today. He used themethod of exhaustion to calculate thearea under the arc of aparabola with the summation of an infinite series,[5] and gave a remarkably accurate approximation ofπ.[80][81]

Mathematicians from theKerala school were studying infinite seriesc. 1350 CE.[82]

In the 17th century,James Gregory worked in the newdecimal system on infinite series and published severalMaclaurin series. In 1715, a general method for constructing theTaylor series for all functions for which they exist was provided byBrook Taylor.Leonhard Euler in the 18th century, developed the theory ofhypergeometric series andq-series.

Convergence criteria

[edit]

The investigation of the validity of infinite series is considered to begin withGauss in the 19th century. Euler had already considered the hypergeometric series

1+αβ1⋅γx+α(α+1)β(β+1)1⋅2⋅γ(γ+1)x2+⋯{\displaystyle 1+{\frac {\alpha \beta }{1\cdot \gamma }}x+{\frac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}x^{2}+\cdots }

on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.

Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The termsconvergence anddivergence had been introduced long before byGregory (1668).Leonhard Euler andGauss had given various criteria, andColin Maclaurin had anticipated some of Cauchy's discoveries. Cauchy advanced the theory ofpower series by his expansion of a complexfunction in such a form.

Abel (1826) in his memoir on thebinomial series

1+m1!x+m(m−1)2!x2+⋯{\displaystyle 1+{\frac {m}{1!}}x+{\frac {m(m-1)}{2!}}x^{2}+\cdots }

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values ofm{\displaystyle m} andx{\displaystyle x}. He showed the necessity of considering the subject of continuity in questions of convergence.

Cauchy's methods led to special rather than general criteria, andthe same may be said ofRaabe (1832), who made the first elaborate investigation of the subject, ofDe Morgan (from 1842), whoselogarithmic testDuBois-Reymond (1873) andPringsheim (1889) haveshown to fail within a certain region; ofBertrand (1842),Bonnet(1843),Malmsten (1846, 1847, the latter without integration);Stokes (1847),Paucker (1852),Chebyshev (1852), andArndt(1853).

General criteria began withKummer (1835), and have been studied byEisenstein (1847),Weierstrass in his variouscontributions to the theory of functions,Dini (1867),DuBois-Reymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.

Uniform convergence

[edit]

The theory ofuniform convergence was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack itsuccessfully wereSeidel andStokes (1847–48). Cauchy took up theproblem again (1853), acknowledging Abel's criticism, and reachingthe same conclusions which Stokes had already found. Thomae used thedoctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniformconvergence, in spite of the demands of the theory of functions.

Semi-convergence

[edit]

A series is said to be semi-convergent (or conditionally convergent) if it is convergent but notabsolutely convergent.

Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, byMalmsten (1847).Schlömilch (Zeitschrift, Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder andBernoulli's function

F(x)=1n+2n+⋯+(x−1)n.{\displaystyle F(x)=1^{n}+2^{n}+\cdots +(x-1)^{n}.}

Genocchi (1852) has further contributed to the theory.

Among the early writers wasWronski, whose "loi suprême" (1815) was hardly recognized untilCayley (1873) brought it intoprominence.

Fourier series

[edit]

Fourier series were being investigatedas the result of physical considerations at the same time thatGauss, Abel, and Cauchy were working out the theory of infiniteseries. Series for the expansion of sines and cosines, of multiplearcs in powers of the sine and cosine of the arc had been treated byJacob Bernoulli (1702) and his brotherJohann Bernoulli (1701) and stillearlier byVieta. Euler andLagrange simplified the subject,as didPoinsot,Schröter,Glaisher, andKummer.

Fourier (1807) set for himself a different problem, toexpand a given function of⁠x{\displaystyle x}⁠ in terms of the sines or cosines ofmultiples of⁠x{\displaystyle x}⁠, a problem which he embodied in hisThéorie analytique de la chaleur (1822). Euler had already given the formulas for determining the coefficients in the series;Fourier was the first to assert and attempt to prove the generaltheorem.Poisson (1820–23) also attacked the problem from adifferent standpoint. Fourier did not, however, settle the questionof convergence of his series, a matter left forCauchy (1826) toattempt and for Dirichlet (1829) to handle in a thoroughlyscientific manner (seeconvergence of Fourier series). Dirichlet's treatment (Crelle, 1829), of trigonometric series was the subject of criticism and improvement byRiemann (1854), Heine,Lipschitz,Schläfli, anddu Bois-Reymond. Among other prominent contributors to the theory oftrigonometric and Fourier series wereDini,Hermite,Halphen,Krause, Byerly andAppell.

Summations over general index sets

[edit]

Definitions may be given for infinitary sums over an arbitrary index setI.{\displaystyle I.}[83] This generalization introduces two main differences from the usual notion of series: first, there may be no specific order given on the setI{\displaystyle I}; second, the setI{\displaystyle I} may be uncountable. The notions of convergence need to be reconsidered for these, then, because for instance the concept ofconditional convergence depends on the ordering of the index set.

Ifa:I↦G{\displaystyle a:I\mapsto G} is afunction from anindex setI{\displaystyle I} to a setG,{\displaystyle G,} then the "series" associated toa{\displaystyle a} is theformal sum of the elementsa(x)∈G{\displaystyle a(x)\in G} over the index elementsx∈I{\displaystyle x\in I} denoted by the

∑x∈Ia(x).{\displaystyle \sum _{x\in I}a(x).}

When the index set is the natural numbersI=N,{\displaystyle I=\mathbb {N} ,} the functiona:N↦G{\displaystyle a:\mathbb {N} \mapsto G} is asequence denoted bya(n)=an.{\displaystyle a(n)=a_{n}.} A series indexed on the natural numbers is an ordered formal sum and so we rewrite∑n∈N{\textstyle \sum _{n\in \mathbb {N} }} as∑n=0∞{\textstyle \sum _{n=0}^{\infty }} in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers

∑n=0∞an=a0+a1+a2+⋯.{\displaystyle \sum _{n=0}^{\infty }a_{n}=a_{0}+a_{1}+a_{2}+\cdots .}

Families of non-negative numbers

[edit]

When summing a family{ai:i∈I}{\displaystyle \left\{a_{i}:i\in I\right\}} of non-negative real numbers over the index setI{\displaystyle I}, define

∑i∈Iai=sup{∑i∈Aai:A⊆I,A finite}∈[0,+∞].{\displaystyle \sum _{i\in I}a_{i}=\sup {\biggl \{}\sum _{i\in A}a_{i}\,:A\subseteq I,A{\text{ finite}}{\biggr \}}\in [0,+\infty ].}

Any sum over non-negative reals can be understood as the integral of a non-negative function with respect to thecounting measure, which accounts for the many similarities between the two constructions.

When the supremum is finite then the set ofi∈I{\displaystyle i\in I} such thatai>0{\displaystyle a_{i}>0} is countable. Indeed, for everyn≥1,{\displaystyle n\geq 1,} thecardinality|An|{\displaystyle \left|A_{n}\right|} of the setAn={i∈I:ai>1/n}{\displaystyle A_{n}=\left\{i\in I:a_{i}>1/n\right\}} is finite because

1n|An|=∑i∈An1n≤∑i∈Anai≤∑i∈Iai<∞.{\displaystyle {\frac {1}{n}}\,\left|A_{n}\right|=\sum _{i\in A_{n}}{\frac {1}{n}}\leq \sum _{i\in A_{n}}a_{i}\leq \sum _{i\in I}a_{i}<\infty .}

Hence the setA={i∈I:ai>0}=⋃n=1∞An{\displaystyle A=\left\{i\in I:a_{i}>0\right\}=\bigcup _{n=1}^{\infty }A_{n}} iscountable.

IfI{\displaystyle I} is countably infinite and enumerated asI={i0,i1,…}{\displaystyle I=\left\{i_{0},i_{1},\ldots \right\}} then the above defined sum satisfies

∑i∈Iai=∑k=0∞aik,{\displaystyle \sum _{i\in I}a_{i}=\sum _{k=0}^{\infty }a_{i_{k}},}provided the value∞{\displaystyle \infty } is allowed for the sum of the series.

Abelian topological groups

[edit]

Leta:I→X{\displaystyle a:I\to X} be a map, also denoted by(ai)i∈I,{\displaystyle \left(a_{i}\right)_{i\in I},} from some non-empty setI{\displaystyle I} into aHausdorffabeliantopological groupX.{\displaystyle X.} LetFinite⁡(I){\displaystyle \operatorname {Finite} (I)} be the collection of allfinitesubsets ofI,{\displaystyle I,} withFinite⁡(I){\displaystyle \operatorname {Finite} (I)} viewed as adirected set,ordered underinclusion⊆{\displaystyle \,\subseteq \,} withunion asjoin. The family(ai)i∈I,{\displaystyle \left(a_{i}\right)_{i\in I},} is said to beunconditionally summable if the followinglimit, which is denoted by∑i∈Iai{\displaystyle \textstyle \sum _{i\in I}a_{i}} and is called thesum of(ai)i∈I,{\displaystyle \left(a_{i}\right)_{i\in I},} exists inX:{\displaystyle X:}

∑i∈Iai:=limA∈Finite⁡(I) ∑i∈Aai=lim{∑i∈Aai:A⊆I,A finite }{\displaystyle \sum _{i\in I}a_{i}:=\lim _{A\in \operatorname {Finite} (I)}\ \sum _{i\in A}a_{i}=\lim {\biggl \{}\sum _{i\in A}a_{i}\,:A\subseteq I,A{\text{ finite }}{\biggr \}}}Saying that the sumS:=∑i∈Iai{\displaystyle \textstyle S:=\sum _{i\in I}a_{i}} is the limit of finite partial sums means that for every neighborhoodV{\displaystyle V} of the origin inX,{\displaystyle X,} there exists a finite subsetA0{\displaystyle A_{0}} ofI{\displaystyle I} such that

S−∑i∈Aai∈V for every finite supersetA⊇A0.{\displaystyle S-\sum _{i\in A}a_{i}\in V\qquad {\text{ for every finite superset}}\;A\supseteq A_{0}.}

BecauseFinite⁡(I){\displaystyle \operatorname {Finite} (I)} is nottotally ordered, this is not alimit of a sequence of partial sums, but rather of anet.[84][85]

For every neighborhoodW{\displaystyle W} of the origin inX,{\displaystyle X,} there is a smaller neighborhoodV{\displaystyle V} such thatV−V⊆W.{\displaystyle V-V\subseteq W.} It follows that the finite partial sums of an unconditionally summable family(ai)i∈I,{\displaystyle \left(a_{i}\right)_{i\in I},} form aCauchy net, that is, for every neighborhoodW{\displaystyle W} of the origin inX,{\displaystyle X,} there exists a finite subsetA0{\displaystyle A_{0}} ofI{\displaystyle I} such that

∑i∈A1ai−∑i∈A2ai∈W for all finite supersets A1,A2⊇A0,{\displaystyle \sum _{i\in A_{1}}a_{i}-\sum _{i\in A_{2}}a_{i}\in W\qquad {\text{ for all finite supersets }}\;A_{1},A_{2}\supseteq A_{0},}which implies thatai∈W{\displaystyle a_{i}\in W} for everyi∈I∖A0{\displaystyle i\in I\setminus A_{0}} (by takingA1:=A0∪{i}{\displaystyle A_{1}:=A_{0}\cup \{i\}} andA2:=A0{\displaystyle A_{2}:=A_{0}}).

WhenX{\displaystyle X} iscomplete, a family(ai)i∈I{\displaystyle \left(a_{i}\right)_{i\in I}} is unconditionally summable inX{\displaystyle X} if and only if the finite sums satisfy the latter Cauchy net condition. WhenX{\displaystyle X} is complete and(ai)i∈I,{\displaystyle \left(a_{i}\right)_{i\in I},} is unconditionally summable inX,{\displaystyle X,} then for every subsetJ⊆I,{\displaystyle J\subseteq I,} the corresponding subfamily(aj)j∈J,{\displaystyle \left(a_{j}\right)_{j\in J},} is also unconditionally summable inX.{\displaystyle X.}

When the sum of a family of non-negative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological groupX=R.{\displaystyle X=\mathbb {R} .}

If a family(ai)i∈I{\displaystyle \left(a_{i}\right)_{i\in I}} inX{\displaystyle X} is unconditionally summable then for every neighborhoodW{\displaystyle W} of the origin inX,{\displaystyle X,} there is a finite subsetA0⊆I{\displaystyle A_{0}\subseteq I} such thatai∈W{\displaystyle a_{i}\in W} for every indexi{\displaystyle i} not inA0.{\displaystyle A_{0}.} IfX{\displaystyle X} is afirst-countable space then it follows that the set ofi∈I{\displaystyle i\in I} such thatai≠0{\displaystyle a_{i}\neq 0} is countable. This need not be true in a general abelian topological group (see examples below).

Unconditionally convergent series

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Suppose thatI=N.{\displaystyle I=\mathbb {N} .} If a familyan,n∈N,{\displaystyle a_{n},n\in \mathbb {N} ,} is unconditionally summable in a Hausdorffabelian topological groupX,{\displaystyle X,} then the series in the usual sense converges and has the same sum,

∑n=0∞an=∑n∈Nan.{\displaystyle \sum _{n=0}^{\infty }a_{n}=\sum _{n\in \mathbb {N} }a_{n}.}

By nature, the definition of unconditional summability is insensitive to the order of the summation. When∑an{\displaystyle \textstyle \sum a_{n}} is unconditionally summable, then the series remains convergent after anypermutationσ:N→N{\displaystyle \sigma :\mathbb {N} \to \mathbb {N} } of the setN{\displaystyle \mathbb {N} } of indices, with the same sum,

∑n=0∞aσ(n)=∑n=0∞an.{\displaystyle \sum _{n=0}^{\infty }a_{\sigma (n)}=\sum _{n=0}^{\infty }a_{n}.}

Conversely, if every permutation of a series∑an{\displaystyle \textstyle \sum a_{n}} converges, then the series is unconditionally convergent. WhenX{\displaystyle X} iscomplete then unconditional convergence is also equivalent to the fact that all subseries are convergent; ifX{\displaystyle X} is aBanach space, this is equivalent to say that for every sequence of signsεn=±1{\displaystyle \varepsilon _{n}=\pm 1}, the series

∑n=0∞εnan{\displaystyle \sum _{n=0}^{\infty }\varepsilon _{n}a_{n}}

converges inX.{\displaystyle X.}

Series in topological vector spaces

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IfX{\displaystyle X} is atopological vector space (TVS) and(xi)i∈I{\displaystyle \left(x_{i}\right)_{i\in I}} is a (possiblyuncountable) family inX{\displaystyle X} then this family issummable[86] if the limitlimA∈Finite⁡(I)xA{\displaystyle \textstyle \lim _{A\in \operatorname {Finite} (I)}x_{A}} of thenet(xA)A∈Finite⁡(I){\displaystyle \left(x_{A}\right)_{A\in \operatorname {Finite} (I)}} exists inX,{\displaystyle X,} whereFinite⁡(I){\displaystyle \operatorname {Finite} (I)} is thedirected set of all finite subsets ofI{\displaystyle I} directed by inclusion⊆{\displaystyle \,\subseteq \,} andxA:=∑i∈Axi.{\textstyle x_{A}:=\sum _{i\in A}x_{i}.}

It is calledabsolutely summable if in addition, for every continuous seminormp{\displaystyle p} onX,{\displaystyle X,} the family(p(xi))i∈I{\displaystyle \left(p\left(x_{i}\right)\right)_{i\in I}} is summable.IfX{\displaystyle X} is a normable space and if(xi)i∈I{\displaystyle \left(x_{i}\right)_{i\in I}} is an absolutely summable family inX,{\displaystyle X,} then necessarily all but a countable collection ofxi{\displaystyle x_{i}}’s are zero. Hence, in normed spaces, it is usually only ever necessary to consider series with countably many terms.

Summable families play an important role in the theory ofnuclear spaces.

Series in Banach and seminormed spaces

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The notion of series can be easily extended to the case of aseminormed space. Ifxn{\displaystyle x_{n}} is a sequence of elements of a normed spaceX{\displaystyle X} and ifx∈X{\displaystyle x\in X} then the series∑xn{\displaystyle \textstyle \sum x_{n}} converges tox{\displaystyle x} inX{\displaystyle X} if the sequence of partial sums of the series( ∑n=0Nxn)N=1∞{\textstyle {\bigl (}\!\!~\sum _{n=0}^{N}x_{n}{\bigr )}_{N=1}^{\infty }} converges tox{\displaystyle x} inX{\displaystyle X}; to wit,

‖x−∑n=0Nxn‖→0 as N→∞.{\displaystyle {\Biggl \|}x-\sum _{n=0}^{N}x_{n}{\Biggr \|}\to 0\quad {\text{ as }}N\to \infty .}

More generally, convergence of series can be defined in anyabelianHausdorfftopological group. Specifically, in this case,∑xn{\displaystyle \textstyle \sum x_{n}} converges tox{\displaystyle x} if the sequence of partial sums converges tox.{\displaystyle x.}

If(X,|⋅|){\displaystyle (X,|\cdot |)} is aseminormed space, then the notion of absolute convergence becomes: A series∑i∈Ixi{\textstyle \sum _{i\in I}x_{i}} of vectors inX{\displaystyle X}converges absolutely if

∑i∈I|xi|<+∞{\displaystyle \sum _{i\in I}\left|x_{i}\right|<+\infty }

in which case all but at most countably many of the values|xi|{\displaystyle \left|x_{i}\right|} are necessarily zero.

If a countable series of vectors in a Banach space converges absolutely then it converges unconditionally, but the converse only holds in finite-dimensional Banach spaces (theorem ofDvoretzky & Rogers (1950)).

Well-ordered sums

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Conditionally convergent series can be considered ifI{\displaystyle I} is awell-ordered set, for example, anordinal numberα0.{\displaystyle \alpha _{0}.}In this case, define bytransfinite recursion:

∑β<α+1aβ=aα+∑β<αaβ{\displaystyle \sum _{\beta <\alpha +1}\!a_{\beta }=a_{\alpha }+\sum _{\beta <\alpha }a_{\beta }}

and for a limit ordinalα,{\displaystyle \alpha ,}

∑β<αaβ=limγ→α∑β<γaβ{\displaystyle \sum _{\beta <\alpha }a_{\beta }=\lim _{\gamma \to \alpha }\,\sum _{\beta <\gamma }a_{\beta }}

if this limit exists. If all limits exist up toα0,{\displaystyle \alpha _{0},} then the series converges.

Examples

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See also

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Notes

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  1. ^Thompson, Silvanus;Gardner, Martin (1998).Calculus Made Easy. Macmillan.ISBN 978-0-312-18548-0.
  2. ^Huggett, Nick (2024),"Zeno's Paradoxes", in Zalta, Edward N.; Nodelman, Uri (eds.),The Stanford Encyclopedia of Philosophy (Spring 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved2024-03-25
  3. ^Apostol 1967, pp. 374–375
  4. ^Swain, Gordon; Dence, Thomas (1998)."Archimedes' Quadrature of the Parabola Revisited".Mathematics Magazine.71 (2):123–130.doi:10.2307/2691014.ISSN 0025-570X.JSTOR 2691014.
  5. ^abRusso, Lucio (2004).The Forgotten Revolution. Translated by Levy, Silvio. Germany: Springer-Verlag. pp. 49–52.ISBN 978-3-540-20396-4.
  6. ^Apostol 1967, p. 377
  7. ^Apostol 1967, p. 378
  8. ^abcApostol 1967, p. 37
  9. ^abcdefSpivak 2008, pp. 471–472
  10. ^abcdeApostol 1967, p. 384
  11. ^abcdefAblowitz, Mark J.; Fokas, Athanassios S. (2003).Complex Variables: Introduction and Applications (2nd ed.). Cambridge University Press. p. 110.ISBN 978-0-521-53429-1.
  12. ^abDummit, David S.; Foote, Richard M. (2004).Abstract Algebra (3rd ed.). Hoboken, NJ: John Wiley and Sons. p. 238.ISBN 978-0-471-43334-7.
  13. ^abcdSpivak 2008, pp. 486–487, 493
  14. ^abWilf, Herbert S. (1990).Generatingfunctionology. San Diego: Academic Press. pp. 27–28.ISBN 978-1-48-324857-8.
  15. ^Swokoski, Earl W. (1983).Calculus with Analytic Geometry (Alternate ed.). Boston: Prindle, Weber & Schmidt. p. 501.ISBN 978-0-87150-341-1.
  16. ^abcdRudin 1976, p. 59
  17. ^Spivak 2008, p. 426
  18. ^Apostol 1967, p. 281
  19. ^Rudin 1976, p. 63
  20. ^abcdeSpivak 2008, pp. 473–478
  21. ^abcdeApostol 1967, pp. 388–390, 399–401
  22. ^abcRudin 1976, p. 61
  23. ^Spivak 2008, p. 453
  24. ^Knuth, Donald E. (1992). "Two Notes on Notation".American Mathematical Monthly.99 (5):403–422.doi:10.2307/2325085.JSTOR 2325085.
  25. ^Atkinson, Kendall E. (1989).An Introduction to Numerical Analysis (2nd ed.). New York: Wiley. p. 20.ISBN 978-0-471-62489-9.OCLC 803318878.
  26. ^Stoer, Josef; Bulirsch, Roland (2002).Introduction to Numerical Analysis (3rd ed.). Princeton, N.J.: Recording for the Blind & Dyslexic.OCLC 50556273.
  27. ^Wilkins, David (2007)."Section 6: The Extended Real Number System"(PDF).maths.tcd.ie. Retrieved2019-12-03.
  28. ^abKifowit, Steven J.; Stamps, Terra A. (2006)."The harmonic series diverges again and again"(PDF).American Mathematical Association of Two-Year Colleges Review.27 (2):31–43.
  29. ^abcSpivak 2008, p. 496
  30. ^abRudin 1976, p. 61
  31. ^Spivak 2008, pp. 483–486
  32. ^Apostol 1967, pp. 412–414
  33. ^Rudin 1976, p. 76
  34. ^Spivak 2008, p. 482
  35. ^abcApostol 1967, pp. 385–386
  36. ^abcSaff, E. B.; Snider, Arthur D. (2003).Fundamentals of Complex Analysis (3rd ed.). Pearson Education. pp. 247–249.ISBN 0-13-907874-6.
  37. ^Rudin 1976, p. 72
  38. ^Rudin 1976, p. 73
  39. ^Rudin 1976, p. 74
  40. ^Apostol 1967, p. 384
  41. ^Apostol 1967, pp. 403–404
  42. ^Apostol 1967, p. 386
  43. ^abApostol 1967, p. 387
  44. ^Apostol 1967, p. 396
  45. ^Gasper, G., Rahman, M. (2004). Basic hypergeometric series.Cambridge University Press.
  46. ^Spivak 2008, p. 473
  47. ^abcRudin 1976, p. 60
  48. ^Apostol 1967, pp. 381, 394–395
  49. ^Spivak 2008, pp. 457, 473–474
  50. ^Rudin 1976, pp. 71–72
  51. ^Apostol 1967, pp. 395–396
  52. ^Spivak 2008, pp. 474–475
  53. ^Apostol 1967, p. 396
  54. ^Spivak 2008, p. 475–476
  55. ^abApostol 1967, pp. 399–401
  56. ^Spivak 2008, pp. 476–478
  57. ^Rudin 1976, p. 66
  58. ^Spivak 2008, p. 493
  59. ^Rudin 1976, p. 65
  60. ^Apostol 1967, pp. 397–398
  61. ^Spivak 2008, pp. 478–479
  62. ^Apostol 1967, pp. 403–404
  63. ^Spivak 2008, p. 481
  64. ^Rudin 1976, p. 71
  65. ^Apostol 1967, pp. 413–414
  66. ^Spivak 2008, pp. 482–483
  67. ^abApostol 1967, pp. 407–409
  68. ^Spivak 2008, p. 495
  69. ^Rudin 1976, p. 70
  70. ^Spivak 2008, p. 524
  71. ^Positive and Negative Terms: Alternating Series
  72. ^Johansson, F. (2016). Computing hypergeometric functions rigorously. arXiv preprint arXiv:1606.06977.
  73. ^Higham, N. J. (2008). Functions of matrices: theory and computation.Society for Industrial and Applied Mathematics.
  74. ^Higham, N. J. (2009). The scaling and squaring method for the matrix exponential revisited. SIAM review, 51(4), 747-764.
  75. ^How and How Not to Compute the Exponential of a Matrix
  76. ^Nicolas Bourbaki (1989),Algebra, Springer: §III.2.11.
  77. ^Huggett, Nick (2024),"Zeno's Paradoxes", in Zalta, Edward N.; Nodelman, Uri (eds.),The Stanford Encyclopedia of Philosophy (Spring 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved2024-03-25
  78. ^Snyder, H. (1947), "Quantized space-time",Physical Review,67 (1):38–41,Bibcode:1947PhRv...71...38S,doi:10.1103/PhysRev.71.38.
  79. ^"The Unraveling of Space-Time".Quanta Magazine. 2024-09-25. Retrieved2024-10-11.
  80. ^O'Connor, J.J. & Robertson, E.F. (1996)."A history of calculus".University of St Andrews. Retrieved2007-08-07.
  81. ^Bidwell, James K. (30 November 1993). "Archimedes and Pi-Revisited".School Science and Mathematics.94 (3):127–129.doi:10.1111/j.1949-8594.1994.tb15638.x.
  82. ^"Indians predated Newton 'discovery' by 250 years".manchester.ac.uk.
  83. ^Jean Dieudonné,Foundations of mathematical analysis, Academic Press[page needed]
  84. ^Bourbaki, Nicolas (1998).General Topology: Chapters 1–4. Springer. pp. 261–270.ISBN 978-3-540-64241-1.
  85. ^Choquet, Gustave (1966).Topology. Academic Press. pp. 216–231.ISBN 978-0-12-173450-3.
  86. ^Schaefer, Helmut H.; Wolff, Manfred P. (1999).Topological Vector Spaces. Graduate Texts in Mathematics. Vol. 8 (2nd ed.). New York, NY: Springer. pp. 179–180.ISBN 978-1-4612-7155-0.

References

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Further reading

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External links

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