Movatterモバイル変換

Jump to content

Divergent series

From Wikipedia, the free encyclopedia

Infinite series that is not convergent

For the publication, seeDivergent (book series). For the film series, seeThe Divergent Series.

Les séries divergentes sont en général quelque chose de bien fatal et c’est une honte qu’on ose y fonder aucune démonstration.("Divergent series are in general something fatal, and it is a disgrace to base any proof on them." Often translated as "Divergent series are an invention of the devil …")

N. H. Abel, letter to Holmboe, January 1826, reprinted in volume 2 of his collected papers.

Inmathematics, adivergent series is aninfinite series that is notconvergent, meaning that the infinitesequence of thepartial sums of the series does not have a finitelimit.

If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. Acounterexample is theharmonic series

1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n}}.

The divergence of the harmonic serieswas proven by the medieval mathematicianNicole Oresme.

In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. Asummability method orsummation method is apartial function from the set of series to values. For example,Cesàro summation assignsGrandi's divergent series

1-1+1-1+\cdots

the value⁠1/2⁠. Cesàro summation is anaveraging method, in that it relies on thearithmetic mean of the sequence of partial sums. Other methods involveanalytic continuations of related series. Inphysics, there are a wide variety of summability methods; these are discussed in greater detail in the article onregularization.

History

... but it is broadly true to say that mathematicians before Cauchy asked not 'How shall wedefine 1 − 1 + 1...?' but 'Whatis 1 − 1 + 1...?', and that this habit of mind led them into unnecessary perplexities and controversies which were often really verbal.

G. H. Hardy, Divergent series, page 6

Before the 19th century, divergent series were widely used byLeonhard Euler and others, but often led to confusing and contradictory results. A major problem was Euler's idea that any divergent series should have a natural sum, without first defining what is meant by the sum of a divergent series.Augustin-Louis Cauchy eventually gave a rigorous definition of the sum of a (convergent) series, and for some time after this, divergent series were mostly excluded from mathematics. They reappeared in 1886 withHenri Poincaré's work on asymptotic series. In 1890,Ernesto Cesàro realized that one could give a rigorous definition of the sum of some divergent series, and definedCesàro summation. (This was not the first use of Cesàro summation, which was used implicitly byFerdinand Georg Frobenius in 1880; Cesàro's key contribution was not the discovery of this method, but his idea that one should give an explicit definition of the sum of a divergent series.) In the years after Cesàro's paper, several other mathematicians gave other definitions of the sum of a divergent series, although these are not always compatible: different definitions can give different answers for the sum of the same divergent series; so, when talking about the sum of a divergent series, it is necessary to specify which summation method one is using.

Examples

1 - 1 + 1 - 1 + ⋯ ${\text{ “ = ” }}{\frac {1}{2}}$
1 − 2 + 3 − 4 + ⋯ ${\text{ “ = ” }}{\frac {1}{4}}$
1 − 1 + 2 − 6 + 24 − 120 + ⋯ ${\text{ “ = ” }}\!\!\int _{0}^{\infty }{\frac {e^{-x}}{1+x}}\,dx\approx 0.596\,347\ldots$
1 − 2 + 4 − 8 + ⋯ ${\text{ “ = ” }}{\frac {1}{3}}$
1 + 2 + 4 + 8 + ⋯ ${\text{ “ = ” }}-1$
1 + 1 + 1 + 1 + ⋯ ${\text{ “ = ” }}-{\frac {1}{2}}$
1 + 2 + 3 + 4 + ⋯ ${\text{ “ = ” }}-{\frac {1}{12}}$

Theorems on methods for summing divergent series

A summability methodM isregular if it agrees with the actual limit on allconvergent series. Such a result is called anAbelian theorem forM, from the prototypicalAbel's theorem. More subtle, are partial converse results, calledTauberian theorems, from a prototype proved byAlfred Tauber. Herepartial converse means that ifM sums the seriesΣ, and some side-condition holds, thenΣ was convergent in the first place; without any side-condition such a result would say thatM only summed convergent series (making it useless as a summation method for divergent series).

The function giving the sum of a convergent series islinear, and it follows from theHahn–Banach theorem that it may be extended to a summation method summing any series with bounded partial sums. This is called theBanach limit. This fact is not very useful in practice, since there are many such extensions, inconsistent with each other, and also since proving such operators exist requires invoking theaxiom of choice or its equivalents, such asZorn's lemma. They are therefore nonconstructive.

The subject of divergent series, as a domain ofmathematical analysis, is primarily concerned with explicit and natural techniques such asAbel summation,Cesàro summation andBorel summation, and their relationships. The advent ofWiener's tauberian theorem marked an epoch in the subject, introducing unexpected connections toBanach algebra methods inFourier analysis.

Summation of divergent series is also related toextrapolation methods andsequence transformations as numerical techniques. Examples of such techniques arePadé approximants,Levin-type sequence transformations, and order-dependent mappings related torenormalization techniques for large-orderperturbation theory inquantum mechanics.

Properties of summation methods

Summation methods usually concentrate on the sequence of partial sums of the series. While this sequence does not converge, we may often find that when we take an average of larger and larger numbers of initial terms of the sequence, the average converges, and we can use this average instead of a limit to evaluate the sum of the series. Asummation method can be seen as a function from a set of sequences of partial sums to values. IfA is any summation method assigning values to a set of sequences, we may mechanically translate this to aseries-summation methodA^Σ that assigns the same values to the corresponding series. There are certain properties it is desirable for these methods to possess if they are to arrive at values corresponding to limits and sums, respectively.

Regularity. A summation method isregular if, whenever the sequences converges tox,A(s) =x. Equivalently, the corresponding series-summation method evaluatesA^Σ(a) =x.
Linearity.A islinear if it is a linear functional on the sequences where it is defined, so thatA(kr +s) =kA(r) +A(s) for sequencesr,s and a real or complex scalark. Since the termsa_n+1 =s_n+1 −s_n of the seriesa are linear functionals on the sequences and vice versa, this is equivalent toA^Σ being a linear functional on the terms of the series.
Stability (also calledtranslativity). Ifs is a sequence starting froms₀ ands′ is the sequence obtained by omitting the first value and subtracting it from the rest, so thats′_n =s_n+1 −s₀, thenA(s) is defined if and only ifA(s′) is defined, andA(s) =s₀ +A(s′). Equivalently, whenevera′_n =a_n+1 for alln, thenA^Σ(a) =a₀ +A^Σ(a′).^[1]^[2] Another way of stating this is that theshift rule must be valid for the series that are summable by this method.

The third condition is less important, and some significant methods, such asBorel summation, do not possess it.^[3]

One can also give a weaker alternative to the last condition.

Finite re-indexability. Ifa anda′ are two series such that there exists abijection $f:\mathbb {N} \rightarrow \mathbb {N}$ such thata_i =a′_f(i) for alli, and if there exists some $N\in \mathbb {N}$ such thata_i =a′_i for alli > N, thenA^Σ(a) =A^Σ(a′). (In other words,a′ is the same series asa, with only finitely many terms re-indexed.) This is a weaker condition thanstability, because any summation method that exhibitsstability also exhibitsfinite re-indexability, but the converse is not true.)

A desirable property for two distinct summation methodsA andB to share isconsistency:A andB areconsistent if for every sequences to which both assign a value,A(s) =B(s). (Using this language, a summation methodA is regular iff it is consistent with the standard sumΣ.) If two methods are consistent, and one sums more series than the other, the one summing more series isstronger.

There are powerful numerical summation methods that are neither regular nor linear, for instance nonlinearsequence transformations likeLevin-type sequence transformations andPadé approximants, as well as the order-dependent mappings of perturbative series based onrenormalization techniques.

Taking regularity, linearity and stability as axioms, it is possible to sum many divergent series by elementary algebraic manipulations. This partly explains why many different summation methods give the same answer for certain series.

For instance, wheneverr ≠ 1, thegeometric series

{\begin{aligned}G(r,c)&=\sum _{k=0}^{\infty }cr^{k}&&\\&=c+\sum _{k=0}^{\infty }cr^{k+1}&&{\text{ (stability) }}\\&=c+r\sum _{k=0}^{\infty }cr^{k}&&{\text{ (linearity) }}\\&=c+r\,G(r,c),&&{\text{ hence }}\\G(r,c)&={\frac {c}{1-r}},{\text{ unless it is infinite}}&&\\\end{aligned}}

can be evaluated regardless of convergence. More rigorously, any summation method that possesses these properties and which assigns a finite value to the geometric series must assign this value. However, whenr is a real number larger than 1, the partial sums increase without bound, and averaging methods assign a limit of infinity.

Classical summation methods

The two classical summation methods for series, ordinary convergence and absolute convergence, define the sum as a limit of certain partial sums. These are included only for completeness; strictly speaking they are not true summation methods for divergent series since, by definition, a series is divergent only if these methods do not work. Most but not all summation methods for divergent series extend these methods to a larger class of sequences.

Absolute convergence

Absolute convergence defines the sum of a sequence (or set) of numbers to be the limit of the net of all partial sumsa_k₁ + ... +a_{k_n}, if it exists. It does not depend on the order of the elements of the sequence, and a classical theorem says that a sequence is absolutely convergent if and only if the sequence of absolute values is convergent in the standard sense.

Sum of a series

Cauchy's classical definition of the sum of a seriesa₀ +a₁ + ... defines the sum to be the limit of the sequence of partial sumsa₀ + ... +a_n. This is the default definition of convergence of a sequence.

Nørlund means

Supposep_n is a sequence of positive terms, starting fromp₀. Suppose also that

{\frac {p_{n}}{p_{0}+p_{1}+\cdots +p_{n}}}\rightarrow 0.

If now we transform a sequence s by usingp to give weighted means, setting

t_{m}={\frac {p_{m}s_{0}+p_{m-1}s_{1}+\cdots +p_{0}s_{m}}{p_{0}+p_{1}+\cdots +p_{m}}}

then the limit oft_n asn goes to infinity is an average called theNørlund meanN_p(s).

The Nørlund mean is regular, linear, and stable. Moreover, any two Nørlund means are consistent.

Cesàro summation

The most significant of the Nørlund means are the Cesàro sums. Here, if we define the sequencep^k by

p_{n}^{k}={n+k-1 \choose k-1}

then the Cesàro sumC_k is defined byC_k(s) =N_(p^k)(s). Cesàro sums are Nørlund means ifk ≥ 0, and hence are regular, linear, stable, and consistent.C₀ is ordinary summation, andC₁ is ordinaryCesàro summation. Cesàro sums have the property that ifh >k, thenC_h is stronger thanC_k.

Abelian means

Supposeλ = {λ₀,λ₁,λ₂,...} is a strictly increasing sequence tending towards infinity, and thatλ₀ ≥ 0. Suppose

f(x)=\sum _{n=0}^{\infty }a_{n}e^{-\lambda _{n}x}

converges for all real numbersx > 0. Then theAbelian meanA_λ is defined as

A_{\lambda }(s)=\lim _{x\rightarrow 0^{+}}f(x).

More generally, if the series forf only converges for largex but can be analytically continued to all positive realx, then one can still define the sum of the divergent series by the limit above.

A series of this type is known as a generalizedDirichlet series; in applications to physics, this is known as the method ofheat-kernel regularization.

Abelian means are regular and linear, but not stable and not always consistent between different choices ofλ. However, some special cases are very important summation methods.

Abel summation

See also:Abel's theorem

Ifλ_n =n, then we obtain the method ofAbel summation. Here

f(x)=\sum _{n=0}^{\infty }a_{n}e^{-nx}=\sum _{n=0}^{\infty }a_{n}z^{n},

wherez = exp(−x). Then the limit off(x) asx approaches 0 throughpositive reals is the limit of thepower series forf(z) asz approaches 1 from below through positive reals, and the Abel sumA(s) is defined as

A(s)=\lim _{z\rightarrow 1^{-}}\sum _{n=0}^{\infty }a_{n}z^{n}.

Abel summation is interesting in part because it is consistent with but more powerful thanCesàro summation:A(s) =C_k(s) whenever the latter is defined. The Abel sum is therefore regular, linear, stable, and consistent with Cesàro summation.

Lindelöf summation

Ifλ_n =n log(n), then (indexing from one) we have

f(x)=a_{1}+a_{2}2^{-2x}+a_{3}3^{-3x}+\cdots .

ThenL(s), theLindelöf sum,^[4] is the limit off(x) asx goes to positive zero. The Lindelöf sum is a powerful method when applied to power series among other applications, summing power series in theMittag-Leffler star.

Ifg(z) is analytic in a disk around zero, and hence has aMaclaurin seriesG(z) with a positive radius of convergence, thenL(G(z)) =g(z) in the Mittag-Leffler star. Moreover, convergence tog(z) is uniform on compact subsets of the star.

Analytic continuation

Several summation methods involve taking the value of ananalytic continuation of a function.

Analytic continuation of power series

If Σa_nxⁿ converges for small complexx and can be analytically continued along some path fromx = 0 to the pointx = 1, then the sum of the series can be defined to be the value atx = 1. This value may depend on the choice of path. One of the first examples of potentially different sums for a divergent series, using analytic continuation, was given by Callet,^[5] who observed that if $1\leq m<n$ then

${\frac {1-x^{m}}{1-x^{n}}}={\frac {1+x+\dots +x^{m-1}}{1+x+\dots x^{n-1}}}=1-x^{m}+x^{n}-x^{n+m}+x^{2n}-\dots$

Evaluating at $x=1$ , one gets

$1-1+1-1+\dots ={\frac {m}{n}}.$

However, the gaps in the series are key. For $m=1,n=3$ for example, we actually would get

$1-1+0+1-1+0+1-1+\dots ={\frac {1}{3}}$ , so different sums correspond to different placements of the $0 {\displaystyle 0}$ 's.

Another example of analytic continuation is the divergent alternating series $\sum _{k\geq 0}(-1)^{k+1}{\frac {1}{2k-1}}{\binom {2k}{k}}=1+2-2+4-10+28-84+264-858+2860-9724+\cdots$ which is a sum over products of $\Gamma$ -functions andPochhammer's symbols. Using theduplication formula of the $\Gamma$ -function, it reduces toageneralized hypergeometric series $\ldots =\sum _{k\geq 0}(-4)^{k}{\frac {(-1/2)_{k}}{k!}}={}_{1}F_{0}(-1/2;;-4)={\sqrt {5}}.$

Euler summation

Main article:Euler summation

Euler summation is essentially an explicit form of analytic continuation. If a power series converges for small complexz and can be analytically continued to the open disk with diameter from⁠−1/q + 1⁠ to 1 and is continuous at 1, then its value atq is called the Euler or (E,q) sum of the series Σa_n. Euler used it before analytic continuation was defined in general, and gave explicit formulas for the power series of the analytic continuation.

The operation of Euler summation can be repeated several times, and this is essentially equivalent to taking an analytic continuation of a power series to the point z = 1.

Analytic continuation of Dirichlet series

This method defines the sum of a series to be the value of the analytic continuation of the Dirichlet series

f(s)={\frac {a_{1}}{1^{s}}}+{\frac {a_{2}}{2^{s}}}+{\frac {a_{3}}{3^{s}}}+\cdots

ats = 0, if this exists and is unique. This method is sometimes confused with zeta function regularization.

Ifs = 0 is an isolated singularity, the sum is defined by the constant term of the Laurent series expansion.

Zeta function regularization

Main article:Zeta function regularization

If the series

f(s)={\frac {1}{a_{1}^{s}}}+{\frac {1}{a_{2}^{s}}}+{\frac {1}{a_{3}^{s}}}+\cdots

(for positive values of thea_n) converges for large reals and can beanalytically continued along the real line tos = −1, then its value ats = −1 is called thezeta regularized sum of the seriesa₁ + a₂ + ... Zeta function regularization is nonlinear. In applications, the numbersa_i are sometimes the eigenvalues of a self-adjoint operatorA with compact resolvent, andf(s) is then the trace ofA^−s. For example, ifA has eigenvalues 1, 2, 3, ... thenf(s) is theRiemann zeta function,ζ(s), whose value ats = −1 is −⁠1/12⁠, assigning a value to the divergent series1 + 2 + 3 + 4 + ⋯. Other values ofs can also be used to assign values for the divergent sumsζ(0) = 1 + 1 + 1 + ... = −⁠1/2⁠,ζ(−2) = 1 + 4 + 9 + ... = 0 and in general

\zeta (-s)=\sum _{n=1}^{\infty }n^{s}=1^{s}+2^{s}+3^{s}+\cdots =-{\frac {B_{s+1}}{s+1}}\,,

whereB_k is aBernoulli number.^[6]

Integral function means

IfJ(x) = Σp_nxⁿ is an integral function, then theJ sum of the seriesa₀ + ... is defined to be

\lim _{x\rightarrow \infty }{\frac {\sum _{n}p_{n}(a_{0}+\cdots +a_{n})x^{n}}{\sum _{n}p_{n}x^{n}}},

if this limit exists.

There is a variation of this method where the series forJ has a finite radius of convergencer and diverges atx = r. In this case one defines the sum as above, except taking the limit asx tends tor rather than infinity.

Borel summation

In the special case whenJ(x) = e^x this gives one (weak) form ofBorel summation.

Valiron's method

Valiron's method is a generalization of Borel summation to certain more general integral functionsJ. Valiron showed that under certain conditions it is equivalent to defining the sum of a series as

\lim _{n\rightarrow +\infty }{\sqrt {\frac {H(n)}{2\pi }}}\sum _{h\in Z}e^{-{\frac {1}{2}}h^{2}H(n)}(a_{0}+\cdots +a_{h})

whereH is the second derivative ofG andc(n) = e^−G(n), anda₀ + ... + a_h is to be interpreted as 0 when h < 0.

Moment methods

Suppose thatdμ is a measure on the real line such that all the moments

\mu _{n}=\int x^{n}\,d\mu

are finite. Ifa₀ + a₁ + ... is a series such that

a(x)={\frac {a_{0}x^{0}}{\mu _{0}}}+{\frac {a_{1}x^{1}}{\mu _{1}}}+\cdots

converges for allx in the support ofμ, then the (dμ) sum of the series is defined to be the value of the integral

\int a(x)\,d\mu

if it is defined. (If the numbersμ_n increase too rapidly then they do not uniquely determine the measureμ.)

Borel summation

For example, ifdμ = e^−x dx for positivex and 0 for negativex thenμ_n = n!, and this gives one version ofBorel summation, where the value of a sum is given by

\int _{0}^{\infty }e^{-t}\sum {\frac {a_{n}t^{n}}{n!}}\,dt.

There is a generalization of this depending on a variableα, called the (B′,α) sum, where the sum of a seriesa₀ + ... is defined to be

\int _{0}^{\infty }e^{-t}\sum {\frac {a_{n}t^{n\alpha }}{\Gamma (n\alpha +1)}}\,dt

if this integral exists. A further generalization is to replace the sum under the integral by its analytic continuation from small t.

Miscellaneous methods

BGN hyperreal summation

This summation method works by using an extension to the real numbers known as thehyperreal numbers. Since the hyperreal numbers include distinct infinite values, these numbers can be used to represent the values of divergent series. The key method is to designate a particular infinite value that is being summed, usually $\omega$ , which is used as a unit of infinity. Instead of summing to an arbitrary infinity (as is typically done with $\infty$ ), the BGN method sums to the specific hyperreal infinite value labeled $\omega$ . Therefore, the summations are of the form

\sum _{x=1}^{\omega }f(x)

This allows the usage of standard formulas for finite series such asarithmetic progressions in an infinite context. For instance, using this method, the sum of the progression $1+2+3+\ldots$ is ${\frac {\omega ^{2}}{2}}+{\frac {\omega }{2}}$ , or, using just the most significant infinite hyperreal part, ${\frac {\omega ^{2}}{2}}$ .^[7]

Hausdorff transformations

Hardy (1949, chapter 11).

Hölder summation

Main article:Hölder summation

Hutton's method

In 1812 Hutton introduced a method of summing divergent series by starting with the sequence of partial sums, and repeatedly applying the operation of replacing a sequence s₀, s₁, ... by the sequence of averages⁠s₀ + s₁/2⁠,⁠s₁ + s₂/2⁠, ..., and then taking the limit.^[8]

Ingham summability

The seriesa₁ + ... is called Ingham summable tos if

\lim _{x\rightarrow \infty }\sum _{1\leq n\leq x}a_{n}{\frac {n}{x}}\left[{\frac {x}{n}}\right]=s.

Albert Ingham showed that ifδ is any positive number then (C,−δ) (Cesàro) summability implies Ingham summability, and Ingham summability implies (C,δ) summability.^[9]

Lambert summability

The seriesa₁ + ... is calledLambert summable tos if

\lim _{y\rightarrow 0^{+}}\sum _{n\geq 1}a_{n}{\frac {nye^{-ny}}{1-e^{-ny}}}=s.

If a series is (C,k) (Cesàro) summable for anyk then it is Lambert summable to the same value, and if a series is Lambert summable then it is Abel summable to the same value.^[9]

Le Roy summation

The seriesa₀ + ... is called Le Roy summable tos if^[10]

\lim _{\zeta \rightarrow 1^{-}}\sum _{n}{\frac {\Gamma (1+\zeta n)}{\Gamma (1+n)}}a_{n}=s.

Mittag-Leffler summation

The seriesa₀ + ... is called Mittag-Leffler (M) summable tos if^[10]

\lim _{\delta \rightarrow 0}\sum _{n}{\frac {a_{n}}{\Gamma (1+\delta n)}}=s.

Ramanujan summation

Main article:Ramanujan summation

Ramanujan summation is a method of assigning a value to divergent series used by Ramanujan and based on theEuler–Maclaurin summation formula. The Ramanujan sum of a seriesf(0) +f(1) + ... depends not only on the values off at integers, but also on values of the functionf at non-integral points, so it is not really a summation method in the sense of this article.

Riemann summability

The seriesa₁ + ... is called (R,k) (or Riemann) summable tos if^[11]

\lim _{h\rightarrow 0}\sum _{n}a_{n}\left({\frac {\sin nh}{nh}}\right)^{k}=s.

The seriesa₁ + ... is called R₂ summable tos if

\lim _{h\rightarrow 0}{\frac {2}{\pi }}\sum _{n}{\frac {\sin ^{2}nh}{n^{2}h}}(a_{1}+\cdots +a_{n})=s.

Riesz means

Main article:Riesz mean

Ifλ_n form an increasing sequence of real numbers and

A_{\lambda }(x)=a_{0}+\cdots +a_{n}{\text{ for }}\lambda _{n}<x\leq \lambda _{n+1}

then the Riesz (R,λ,κ) sum of the seriesa₀ + ... is defined to be

\lim _{\omega \rightarrow \infty }{\frac {\kappa }{\omega ^{\kappa }}}\int _{0}^{\omega }A_{\lambda }(x)(\omega -x)^{\kappa -1}\,dx.

Vallée-Poussin summability

The seriesa₁ + ... is called VP (or Vallée-Poussin) summable tos if

\lim _{m\rightarrow \infty }\sum _{k=0}^{m}a_{k}{\frac {[\Gamma (m+1)]^{2}}{\Gamma (m+1-k)\,\Gamma (m+1+k)}}=\lim _{m\rightarrow \infty }\left[a_{0}+a_{1}{\frac {m}{m+1}}+a_{2}{\frac {m(m-1)}{(m+1)(m+2)}}+\cdots \right]=s,

where $\Gamma (x)$ is the gamma function.^[11]

Zeldovich summability

Main article:Zeldovich regularization

The series is Zeldovich summable if

\lim _{\alpha \to 0^{+}}\sum _{n}c_{n}e^{-\alpha n^{2}}=s.

See also

Silverman–Toeplitz theorem

Notes

^"Summation methods".Michon's Numericana.
^"Translativity".The Encyclopedia of Mathematics. Springer.
^Muraev, E. B. (1978), "Borel summation ofn-multiple series, and entire functions associated with them",Akademiya Nauk SSSR,19 (6):1332–1340, 1438,MR 0515185. Muraev observes that Borel summation is translative in one of the two directions: augmenting a series by a zero placed at its start does not change the summability or value of the series. However, he states "the converse is false".
^Volkov 2001.
^Hardy 1949, p. 14.
^Tao, Terence (10 April 2010)."The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation".
^Bartlett, Jonathan; Gaastra, Logan; Nemati, David (January 2020). "Hyperreal Numbers for Infinite Divergent Series".Communications of the Blyth Institute.2 (1):7–15.arXiv:1804.11342.doi:10.33014/issn.2640-5652.2.1.bartlett-et-al.1.S2CID 119665957.
^Hardy 1949, p. 21.
^^a ^bHardy 1949, Appendix II.
^^a ^bHardy 1949, 4.11.
^^a ^bHardy 1949, 4.17.

References

Arteca, G.A.; Fernández, F.M.; Castro, E.A. (1990),Large-Order Perturbation Theory and Summation Methods in Quantum Mechanics, Berlin: Springer-Verlag.
Baker, Jr., G. A.; Graves-Morris, P. (1996),Padé Approximants, Cambridge University Press.
Brezinski, C.;Redivo Zaglia, M. (1991),Extrapolation Methods. Theory and Practice, North-Holland.
Hardy, G. H. (1949),Divergent Series, Oxford: Clarendon Press.
LeGuillou, J.-C.; Zinn-Justin, J. (1990),Large-Order Behaviour of Perturbation Theory, Amsterdam: North-Holland.
Volkov, I.I. (2001) [1994],"Lindelöf summation method",Encyclopedia of Mathematics,EMS Press.
Zakharov, A.A. (2001) [1994],"Abel summation method",Encyclopedia of Mathematics,EMS Press.
"Riesz summation method",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
Werner Balser: "From Divergent Power Series to Analytic Functions", Springer-Verlag, LNM 1582, ISBN 0-387-58268-1 (1994).
William O. Bray and Časlav V. Stanojević(Eds.): "Analysis of Divergence", Springer, ISBN 978-1-4612-7467-4 (1999).
Alexander I. Saichev and Wojbor Woyczynski:"Distributions in the Physical and Engineering Sciences, Volume 1", Chap.8 "Summation of divergent series and integrals", Springer (2018).

v
t
e

Sequences andseries

Integer sequences

Basic	Arithmetic progression Geometric progression Harmonic progression Square number Cubic number Factorial Powers of two Powers of three Powers of 10
Advanced(list)	Complete sequence Fibonacci sequence Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number array

Fibonacci spiral with square sizes up to 34.

Properties of sequences

Properties of series

Series	Alternating Convergent Divergent Telescoping
Convergence	Absolute Conditional Uniform

Explicit series

Convergent	1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/2^s + 1/3^s + ... (Riemann zeta function)
Divergent	1 + 1 + 1 + 1 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) 1 + 2 + 3 + 4 + ⋯ 1 − 2 + 3 − 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ Infinite arithmetic series 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)

Kinds of series

Hypergeometric series

Category

Retrieved from "https://en.wikipedia.org/w/index.php?title=Divergent_series&oldid=1277010107"

Hidden categories:

[8]ページ先頭

©2009-2025 Movatter.jp