All[ Hardy's ] books gave him some degree of pleasure,but this one, his last, was his favourite. [ ... ] He believed in its value (as well he might). Preface, byJohn E. Littlewood, to thelast book of G.H. Hardy(1877-1947): "Divergent Series" (1949)
Being eternal, Logic can be patient.Pierre Duhem (1913)
Analytic continuations can make sense of some divergent series in a consistent way. Consider the following classic summation formula (attributed to Eudoxus) for the geometric series, which converges when the common ratio z is small enough in magnitude (it diverges when |z| > 1 ) :
1 + z + z2 + z3 + z4 + ... + zn + ... = 1/ (1-z)
The right-hand-side always makes sense, unless z = 1. It's tempting to equate it formally to the left-hand-side, even when that series diverges!
This viewpoint belongs to a consistent body of knowledge which is stillnot mature, in spite of its exploration by generations of great mathematicians. The following monstrosities do make sense as "sums" of divergent series :
Inrings, whenever both sides of such equations are defined,they are necessarily equal. In p-adicarithmetic, for example, the above geometric series converges (only) when z is an integerwhich is divisible by the modulus p (use p=2 and p=3, respectively, for the above two examples).
Convergence in some ad hoc realm is acomforting luxury which is rarely available. There's no such thing in the case of what's arguablythe simplest divergent series (Grandi's series, 1703) whose long history started with an incidental remark of Leibniz in 1674 (De Triangulo Harmonico) :
1 1 + 1 1 + ... + (-1)n + ... =
The best way to index the successive terms in Grandi's series is to start with n = 1. That makes it a multiplicative sequence. The correct way to extend any multiplicative sequencedown to the index n = 0 (when needed) is to assign it a value of zero there. Thus, I consider dubious the popular useof A033999 as a representation of Grandi's series. I'm adamant about using instead the followingsequence (whose closed form on the left-hand-side depends on the fact that the zeroth power of zero is unity).
Because Grandi's series is stable (as a geometric series whose ratio isn't unity) the leading zero is irrelevant to the summation. This indexation which makes Grandi's sequence multiplicative also makes its Dirichlet inverse multiplicative (and characterized by values at powers of primes):
That's yet another way to show that Grandi's series has a sum of .
(2018-05-23) The equal sign may be used to say that a scalar is the sum of a series.
A (formal) series is just a vectorial object consisting of an infinite sequence ofscalar coefficients (called terms).
The formal series of general term an can be denoted:
nan
A summation method (also summation mapping or summation, for short) is a recipe which assigns a scalar value (called sum) to some series in such a way that the newly defined sum of a series with only finitely many nonzero terms is the ordinary (finite) sum of thosenonzero terms.
The sum of a series using the summation method M is denoted with the nameof M above the sigma sign (or as a superscript). If the summation method is clear fromthe context, square brackets may also be used (such bracket thus turn a series into a scalar):
[nan ] =
n
an
For typographical simplicity and for compatibility with traditional notations, bracketsmay be dropped in equations when it's clear that summation is intended (because the vectorial object itself wouldn'tmake sense in the context). That's so when a series is equated to an explicit scalar. Example:
[nan ] = nan
In particular, the above requirements for any summation method imply:
0 = n 0
Normally, the index n runs from 0 to infinity, through the set of natural integers = {0,1,2,3,...}. It's sometimes convenient to the use the set of counting numbers instead* = {1,2,3,4,...). When we do so, we signal it with a star after the index (this is not a standard notation). In other words:
n*an = nan+1
That's just an equivalence of notation, without a deep mathematical content: It just says that a sequence of scalars remains the same no matter how we index it. The above just expresses the identity between two ways to denote the same formal vectorial objectirrespective of any property attached to it (including summability). Formal series are just vectors with a (countable) infinityof coordinates. Such vectors are equal if and only if all of their coordinates arepairwise identical, as is the case with only finitely many dimensions. Another notation for that same vectorial identity is:
an =
an+1
On the other hand, if we put anything above thesigma sign, then some kind of summation is implied. This could be via one of the highbrow methods discussed on the present page (or elsewhere) or by straight convergence (when applicable) in which case the infinity symbol () can be used. That way, we retrieve a familiar notation exemplified by:
1/ 2n = 2
Thus, the infinity symbol takes the place we may use to identifyother summation methods which gives new meaning to thenice title of a relevant short video by Henry Reich: Adding Past Infinity.
Reich was just lucky that the divergent example he pickedhappened to be a stable series. If we do away with the above sigma notation (as we often will) the stability property Reich took for granted says that:
a0 + (a1 +a2 +a3 + ... ) = a0 +a1 +a2 +a3 + ...
For some series this isn't so. Such unstable series include some physically relevant ones which motivated Reich's next episode, like:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + ...
That series isn't stable (as we shall see) but we can still uncover a sum for it (namely, -1/12). That sum is modified by adding finitely many zeroes "in front of it", but in a consistent way which relies only on linearity.
To navigate this foggy land, we can only rely on precise instruments...
(2012-07-29) Two consistent methods give the same result when both are defined.
The mapping which assigns a sum to a series ought to possess a few desirable properties, but we must be prepared to abandon some of these (except linearity) to allow the most powerful summation methods (e.g., those which can handle interesting unstable divergent series).
1 - Linearity :
Let's split linearity into two separate requirements :
Scalability :nan = nan
Additivity :n (an +bn ) = nan + nbn
Linear summation methods are thus linear forms defined over the vector space of formal series (in any vector space, a form isgenerally defined as a function which takes a vector to a scalar).
2 - Stability (a.k.a. Translativity or Shift-Invariance ) :
Stability is the following nontrivial rule, inspired by finite sums:
nan = a0 + nan+1
That's also known as shift-invariance (explicitely required for summation by Banach limit, when the sequence of partial sums is almost convergent ).
An uresolved question is whether a shifted stable series is necessarily stable again. If that's so, the above rule says that you can freely shift such a series tothe right or to the left without compromising summabilityand that the sum is only affected as would be naively expected (there's no reason to distinguish between right or left stability forlinear summation methods).
The stability property is treacherous. It turns out that it's really a property of the series itself and not of the summation method being used.
A great deal of confusion comes from the fact thatmany summation methods are valid only for stable series (summation by convergence being the most obviousexample). In such a case (and only in such a case) we say that the summation method itselfis translative.
3 - Regularity :
A summation method is said to be regular if it assignsto any convergent series its ordinarysum (i.e., the limit of its partial sums).
By definition, all summation methods are finitely regular (i.e., they all give the same result for any series with finitely many nonzero terms) but some of those are not regular (: consider the Riemann series theorem).
4 - Multiplicativity :
The formal series form not only a vector space butalso a ring. We'd like the summation ofseries to respect that structure and may want the sumof a Cauchy product toequal the product of the sums of its two factors, namely:
(nan ) (nbn ) = n ( k ≤ nan-kbk )
The ordinary summation ofconvergent series is certainlymultiplicative in that sense. So is the Cesàro-summation ofCesàro-summable series: One example involving divergent (but Cesàro-summable) series is the following pair of series, (the second one is the Cauchy-square of the first):
If the limit of fn (x) is 1 (unity) for every index n, as x tends to x0 in some limiting process,then we have, for the same limiting process:
nan = lim x [ nanfn(x) ]
This may be summarized: The sum of the limits is the limit of the sums (the space of formal series is endowed withtheTychonoff topology).
As every student of Analysis ought to know, that's not so in the restricted realm of convergent series, as the series formed by the limits (of the corresponding terms of in a sequenceof convergent series) is not necessarily a convergent series.
Forregular summation methods,whenever the series involved on the right-hand-side are convergent and their sums have a finite limit,that limit is the same for all factors of convergence (this is left as an exercise for the reader) and can thus be used as a definitionof the sum of the series on the left-hand-side if it happens to be divergent. What the semi-continuity of a summation method ultimately states is that the above equation also holdswhen sums of divergent series are involved on the right-hand-side...
Moore's Theorem (2016-05-05)
The key remark, which I like to call Moore's theorem, is that two different sequences of convergence factors which "work"will give the same result. That's to say that the following equation holds wheneverboth sides make sense with two sequences f and g whose terms all have a limit of 1.
lim x [ nanfn(x) ] = lim x [ nangn(x) ]
In 1900, Alfred Pringsheim defined a doubly-indexed sequence as having a limit L if its terms approach L when both indices tend to infinity.
(2018-05-26) The geometric series whose common ratio is unity cannot be stable.
The geometric series of common ratio 1 is not stable, but all other geometric series can be assigned aprecise stable sum. Let's prove that:
We assume alinear summation method powerful enough to assign a finite sum S to the geometric series at point z. Which is to say:
1 + z + z2 + z3 + z4 + ... + zn + ... = S
Stability would then be expressed by the following relation, which would seem obvious to the uninitiated (it's what's often implicitly used as apreliminary step in the derivation of the sum of convergent geometric series). Actually, that's what we want to prove or disprove :
0 + 1 + z + z2 + z3 + ... + zn + ... = S (?)
Bydefinition, the sum of a serieswith finitely many nonzero terms is easy:
1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + ... = 1
As linearity is assumed, wemay subtract that from our original equation:
0 + z + z2 + z3 + z4 + ... + zn + ... = S-1
Also by linearity, we're allowed to scale that by a factor of 1/z (if z is nonzero) and we obtain the sum of the series we were after:
0 + 1 + z + z2 + z3 + ... + zn + ... = (S-1)/z
So, excluding the trivial case z=0, the stability of the geometric series of common ratio z is equivalent to the equation:
S = (S-1)/ z
This equation has no solution when z=1, which goes to show that the geometric series of common ratio unity is notstable. Otherwise, it implies:
S = 1 / (1-z)
That's the only stable sum a linear summation method can assign to the geometric series of common ratio z.
An ad hoc requirement for linear summation methods could be to impose the stability of geometric series of nonunity ratio. We've refrained from adding that to our basic list. Continuity seems far more fundamental, albeit less elementary, as presented below.
(2020-05-13) Any series whose terms tend to zero is stable.
This applies even when such a series doesn't converge (one example being the harmonic series). This is so because the difference between the seriesand the series obtained by shifting it to the right (by one position or several, with zero padding at the beginning) is a convergent series of zero sum...
Indeed, that difference is just a telescoping series whose partial is equal to a single term of the original series (or the sum of several such terms if we shift by several positions). Since we assume that those terms tend to zero, the limit of the partial sumsis zero; the difference is thus a convergent series of zero sum.
(2018-06-23) It's a stable series when at least one of the factors is stable.
A couple of classical theorems pertain to the convergence and/or summability ofthe Cauchy-product of two series which have prescribed properties. (In all of these cases, the sum of the Cauchy-product is the product of the sums.)
(Cesàro, 1890) If two series areCesàro-summable, so is their Cauchy-product (and the sum of that product is the product of the two sums).
The Cauchy product of two unstable summable series can bea summable series whose sum isn't the product of the two sums (cf. example above).
This doesn't seem to happen when at least one of the two series is stable.
It can be useful to think of the Cauchy product as an additive convolution to be contrasted with the mulplicative convolution corresponding to the Dirichlet product described in the next section. They can be defined using very similarnotations. Compare the following definition of the Cauchy productto the parallel definition given below for the Dirichlet product.
f ×g =
f (d) g (n-d) =
f (i) g (j) =
f
g
In both cases, we may consider what happens whenall series involved are absolutely convergent. In that case, all terms can be freely permuted and the above equality is establishedby remarking that every term f (i) g (j) appears once and only once on either side. In other words, the sum of the product is the product of the sums. That nice statement is not necessarily true in general (we've already seen a counter-example).
Summing by Convolutional Inversion (2019-07-24 )
Over any unital ring, the formal seriesform a unital ring (whose unity isthe series whose only nonzero term is a leading term equal to unity). Any series whose leading term is unity has an inverse with respect to theCauchy product, which is often called its convolutional inverse.
When that inverse is summable and has an invertible sum, it's tempting to believe that the inverse of that is the sum of the original series. In some cases, we can prove that to be true...
(2019-05-15 ) of Two Series The sum of the product is the product of the sums. Or is it?
When all series involved are absolutely convergent, their terms can be freely permuted and the above statement is establishedby remarking that every term f (i) g (j) appears once and only once in either of the following summations:
f *g =
f (d) g (n/d) =
f (i) g (j) =
f
g
Dirichlet Summation Method :
Any series whose first term is nonzero has a Dirichlet-inverse. If that Dirichlet-inverse is summable, then the series is said to be Dirichlet-summable and its sum is the reciprocalof the sum of its Dirichlet-inverse.
That implies a sum of -2 for the Moebius series m(n) whose inverse is:
1 + 1 + 1 + 1 + 1 + ... = -1/2
A less chancy example would be the Dirichlet-inverse of Grandi's series:
... = 2
(2019-07-05 ) Exponentials of infinite series seem less ambiguously defined.
The situation is apparently similar to the way ambiguities are liftedwhen going from the formula of Roger Cotes to Euler's formula. However, the full formalism of Riemann sheets cannot be bypassed in general.
(2018-07-03) Countable additivity yields two distinct ways to obtain such sums.
One special case can be derived from the following termwise equality.
ijaibj = i[jaibj ] = iai [jbj ] = [jbj ]iai
The general relation is guaranteed if we have absolute convergence:
[ijaij ] = [jiaij ]
Otherwise, that only gives a questionable hint. One such hint would be that the series whose n-th term is 0(n), thenumber of divisors of n, has a sum of ¼ (: All decimated series S0(k.k-1) have the same sum).
Likewise for any other divisor functions , where s (n) is defined as the sum of the s-th powers of all the positive divisors of n:
The following example shows that this cannot be applied blindly. It's useful to investigate conditions (besides absolute convergence) which allow this.
-1
+
1/2
+
1/22
+
1/23
+
1/24
+
1/25
+
...
=
0
0
+
-1
+
1/2
+
1/22
+
1/23
+
1/24
+
...
=
0
0
+
0
+
-1
+
1/2
+
1/22
+
1/23
+
...
=
0
0
+
0
+
0
+
-1
+
1/2
+
1/22
+
...
=
0
0
+
0
+
0
+
0
+
-1
+
1/2
+
...
=
0
0
+
0
+
0
+
0
+
0
+
-1
+
...
=
0
...
-1
1/2
1/22
1/23
1/24
1/25
...
=
-2
(2018-07-16) (c.1689) Also in the posthumous legacy of Seki (1712).
Many aspiring algebraists, like my younger self, have rediscovered thosein the elementary context of Faulhaber's formula. Bernouilli numbers are often found in Number Theory andplay a key rôle for infinite series via the Euler-Maclaurin formula and Ramanujan summation. They're best defined through their (exponential) generating function :
z / (1ez) = n Bn zn / n! (convergent for | z | < 2 )
z / (1ez) z / (ez 1) = z
Whenever possible, it's probably best not to take either convention for granted andgive explicitly the first terms of an expansion followed by a general expressioninvolving only Bernoulli numbers of even rank (seeexample below).
(2012-08-03) (In infinitely many dimensions, linear forms could bediscontinuous.)
The formal series (irrespective of their convergences) form a sequence space. UsingDirac's notation,a formal series can be described as a ket :
nan = nan | n > = | >
A continuous and linear summation method is a bra < | Such a bra is a member of the continuous dual of the aforementioned sequence space (the algebraic dual consisting of all linear forms is much larger, by theAxiom of Choice).
Formally, a regular summation can only beequal to the following bra :
< | = n < n |
However, that expression is of no practical value,unlike some of the following methods which describe < | better.
Let's define the (non-invertible) shiftoperatorÛ via:
Duality: The smaller the subspace, the bigger its dual.
In general, the [continuous] dual of a [topological] vector space consists of all [continuous] linear functions defined on it (two such functions being equal if and only if they have the same value everywhere).
The dual of any linear space E so defined is a well-defined set (a subset ofthe Cartesian squareE2 ) which is itself a linear space.
In this work we consider only continuous duals and denote E* the [continuous] dual of E.
Furthermore, our attention is restricted to the case when E is some subspace of the sequences of scalars.
(2012-08-10) The only method Cauchy (1789-1857) would ever recognize.
The sequence of the partial sums of a series is thesequence whose term of index n (usually starting at n = 0) is obtained by adding the finitely many terms whose index in theseries does not exceed n.
If that sequence of partial sums converges to a limit S, the series itself is said to be convergent (of sum S).
For convergent series (at least) we make no typographical distinction between a seriesand its sum. Thus, we express the above as:
nan
=
nan
=
an
=
an
lim
The last equation merely expresses the conventional notation for the quoted limitof partial sums. Nothing else.
When that limit doesn't exist, Cauchy argued that the leftmost expressions don't make sense. Two generations before him, the great Euler had taken the opposite view, rather freely, with great success (Ramanujan would do the same much later). Cauchy simply had deep concerns that the lack of explicit rulesfor manipulating divergent series made any argument based on them doubtful at best. So he decided to rule them out!
The pendulum has swung back. Nowadays, divergent series can be handledwith complete analytical rigor. Both Cauchy and Euler would be pleased...
The following sections will trace the historical path awayfrom Cauchy's strict point of view, then break free completely...
Summation by convergence is just the simplest regular summation method,among mutually consistent ones which apply to divergent series as well.
The other such methods, including those described below, can be classified intotwo broad groups:
Summations by means.
Summations by convergence factors.
Sometimes, those two types of approaches are known to be equivalent.
(2012-09-09) How to extend a continuous summation method to a larger domain.
By definition, an absolutely regular summation method isa continuous functional defined for every absolutely convergent serieswhich coincides with the ordinarysummation by convergence. An absolutely regular summation method need not beregular.
(2012-08-09) The earliest method of summation by convergence factors.
Euler introduced the following definition (expressed in our notations):
nan
=
lim
anxn
1-
Apparently, this is the first explicit use of the postulatedcontinuity of summation. For example, Euler famously derived the following sum:
1 2 + 3 4 + 5 6 + ... + (-1)n(n+1) + ... =
As the limiting case of thisconvergent summation, as z tends to 1 on [0,1[
1 2 z + 3 z2 4 z3 + ... + (-1)n+1 (n+1) z n + ... = 1/ (1+z)2
Note, however, that Euler's method isn't powerful enough to handle the nonalternatingcase (corresponding to z = -1) which is one of our favoriteexamples of an unstable series. This much can be construed as a consequence of the following theorem:
This is a straight consequence of the stabilityof convergent series (as implied by the notation we used for Euler's above definition, only convergent series appear in the right-hand-side). Indeed, the limit of the sum of two sequences is the sum of theirrespective limits, when both exist.
Therefore, the following is a trivial equality between identical series :
nan = n0an tn/n! e-t dt
In 1899, Emile Borel (1871-1956) thus proposed to define the left-hand-side (which could be a divergent series) by equating it to the right-hand-side of the following formula,at least when the new series that is so formedis a power series of t with an infiniteradius of convergence, which makesthe resulting (improper) integral converge:
nan = 0(nan tn/n! ) e-t dt
The bracketed series on the right-hand-side clearly stands a better chanceof converging than the series on the left-hand-side.
For example, in the case of thegeometric series (an = zn ) the above integrandbecomes exp [(z-1)t] which makes the integralon the right-hand-side converge when the real part of z is less than one. We thus have convergence of the Borel formulafor half the plane, whereas the left-hand-side merely converges on adisk of unit radius.
Borel summation, however, is best understood as away to obtain the sum of a series from the sum of another,even if the latter is not convergent...
For example, armed with the formula for the sum of ageometric series (convergent of not) we can use the Borel summation tomake mincemeat of the following series which Euler (E247, 1746) called hypergeometric series of Wallis (the name is obsolete; I recommend the unambiguous name Euler-Gompertz series). He evaluated it in half a dozendistinct consistent ways:
(2012-08-21) All Nørlund means are consistent with one another.
Following Niels E. Nørlund (1919) let's consider an infinite sequence ofcomplex ponderation coefficients, the first of which being nonzero:
c0 0 , c1 , c2 ,c3 , c4 , ... , cn , ...
Let's call C the sequence of thepartial sums of the corresponding series:
Cn = c0 + c1 + c2 +c3 + c4 + ... + cn
We impose the so-called regularity condition :
The positive series of term | cn/ Cn | is convergent.
For any series an with partial sums A n = a0 + ... + an we define:
A'n = ( c0 A n + c1 A n-1 + c2 A n-2 + ... + cn A 0)/ C n
This expression is called a Nørlund mean. If A'n tends to a limit S as n tends to infinity,then S is called the Nørlund-sum of the series an or, equivalently, the Nørlund-limit of the sequence A n .
Remarkably, the value of S doesn't depend on the choice of the sequence of coefficients (with the above regularity restrictions).
(2012-08-26) Consistent with Nørlund summation.
The question of the consistency of Nørlund andHausdorff methods was raised by E. Ullirich and it was answered (in the affirmative) by W. Jurkat and A. Peyerimhoff, in 1954.
(2012-08-09) ()
The divergent series arethe invention of the devil, and it is a shame to base on them any demonstration... Niels H. Abel (1802-1829)
The summation method proposed by Abel is:
(2012-08-09)
(2012-08-28)
As he was reflecting on the summation of thegeometric seriesin 1908, Mittag-Leffler proposed the mostwidely applicable definition he could think of, at the time (using the Gamma function) :
nan =
n
an
(1+n)
lim
+
For the geometric case (an = zn) the right-hand-side converges except when z is a real greater than or equal to 1.
More generally, convergence normally occurs on a Mittag-Leffler star consisting of all points of the complex plane, except the shadows of singular points (i.e., wherever a singular point exists on the straight ray from zero to that point).
(2021-07-27)
Georges Valiron (1884-1955) was notably the thesis advisor of Laurent Schwartz (1915-2002).
(2016-05-08) Generalizations and early attempts at defining the sum of divergent series.
In March 1918 at the University of Washington, Lloyd Leroy Smail(1888-1955) defined the sum of a series by the following equation,whenever the right-hand-side makes sense for a sequence of functions fn (m,x) which all tend to 1 as m tends to infinity and x tends to a given limit point x0 :
an =
lim
lim
an fn (m, x)
He pointed out that many previously devised summation methods merelyreduce to different choices for f in the work of different authors,including:
This applies not only to formal power series outside of their disk of convergence but also when each termof the series is an analytic function of z indexed by n (the zeta series being the prime example of that).
(2018-07-28)
(2012-07-28) A summation method which makes mincemeat of some unstable series.
If f is subexponential and the real part of z is large enough, then the series converges. Otherwise we can extend it by analytic continuation (as is normally done to define thezeta function itself) to obtain the desired sum as the value of F(0) whenever it makes sense that way.
This is well-defined because of the uniqueness of Analytic Continuation. The linearity can be established by noticing that the mapfrom a function to its Dirichlet series is a linear oneand analytic continuation preserves linearity at every point where it makessense, including z=0 supposedly.
We've already noted the unstability of the first example. The unstability of the second one results from the aforementionedlinearity of this summation method. Indeed, by adding or subtracting both sides of the above pair of equations, we obtain:
(2016-04-27) Applying linear summation methods to unstable series.
In the previous section, we saw examples of serieswhose lack of stability could be demonstrated using the linearity of summation. I'm not prepared to abandon the linearity requirement for summation methods,even if that means giving up the "obvious" stability property. In this section, I'll show some of the consequences in more details. For shock value.
Let's revisit the previous argument with the following notations. In the first line, we explicitly invoke the assumptionthat a finite sum is equivalent to a series ending withinfinitely many zeroes.
We could subtract a finite sum from either one of those to obtain the sumof a 2-decimated series, starting with an odd or an even number of zeroes. (This would result in a seamless formula, which we're about to generalize.)
This series is convergent for |z| < 1. It's divergent but stable everywhere else in the complex plane, except at the poleof the right-hand-side (z=1) where the series is unstable of sum -½ (obtained byzeta regularization).
We'll use that series when z is a primitive k-th root of unity to obtain, bylinearity alone, the sum of the decimated series where only every k-th term subsists (we demonstrate the technique elsewhere for convergent series).
Our preliminary example was for k=2 (the only primitive square root of unity is z = -1). Let's do it again for k=3, with z = = exp (2i/3)
3 = 1 1 + + 2 = 0
The first 3 lines below correspond to z = 1, and2. The other ones are just linear combinations thereof, using the coefficients highlighted at right.
1 1 1
+ + +
1 2
+ + +
1 2
+ + +
1 1 1
+ + +
1 2
+ + +
1 2
+ + +
1 1 1
+ ... = -1 / 2 + ... = 1 / (1-) +... = 1 / (1-2)
1 1 1
1 2
1 2
3
+
0
+
0
+
3
+
0
+
0
+
3
+ ... = 1 / 2
0
+
3
+
0
+
0
+
3
+
0
+
0
+ ... = -1 / 2
0
+
0
+
3
+
0
+
0
+
3
+
0
+ ... = -3 / 2
The reader is encouraged to work out the above left-hand sides (mentally) and right-hand sides (with pencil and paper).
Check that adding our three results gives just three times the top equation:
3 + 3 + 3 + 3 + 3 + 3 + 3 + ... = -3 / 2
Let's repeat the above pattern for k=4 with = exp (2i/4)
More generally, the result of rank m consists of k times the sum S0(k,m) of the series formed by retaining the term of rank m and every k-th term thereafter (0 ≤ m < k). It's obtained with = exp (2i/k) by assigning a coefficient -m q to the equation of rank q. That yields:
k S0(k,m) = -1/2 +
- m q/ (1- q)
When all is said and done S0(k,k-1) is always equal to -1/2.
S0(k,m) = 1/2 (m+1)/ k for 0 ≤ m < k
The validity of this formula extends to larger values of m, since subtractingone unit from the leading term of such a k-decimated series clearly lowers the sum by one. But that can also be construed as increasing m by k units. To summarize:
The method and the notations just exemplified for the geometric series apply unchanged tothe decimation (0 ≤ m < k) of any summable power series, convergent or not (: multiply each of the above columns into its own coefficient). Calling f (z) the undecimated sum, we obtain:
Decimation Formula [f is analytic on the unit circle except at 1. f (0) 0 ]
Sf(k,m) =
1
k
exp (- 2i mq/k ) f ( exp (2i q/k) )
Let's use that formula to decimate the series 1 + 2 + 3 + 4 + 5 + ... knowing that n* n zn-1 = f (z) is -1/12 if z=1 and 1/(1-z)2 otherwise.
S1
m=0
m=1
m=2
m=3
m=4
m=5
m=6
m=7
m=8
m=9
k=1
-1/12
-13/12
-37/12
-73/12
-1/12 - m (m+1) / 2
k=2
1/12
-1/6
-11/12
-13/6
-47/12
-37/6
1/12 - m2 / 4
k=3
1/12
1/12
-1/4
-11/12
-23/12
-13/4
1/12 - m (m-1) / 6
k=4
1/24
1/6
1/24
-1/3
-23/24
-11/6
1/24 - m (m-2) / 8
k=5
-1/60
11/60
11/60
-1/60
-5/12
-61/60
-1/60 - m (m-3) / 10
k=6
-1/12
1/6
1/4
1/6
-1/12
-1/2
k=7
-13/84
11/84
23/84
23/84
11/84
-13/84
-7/12
k=8
-11/48
1/12
13/48
1/3
13/48
1/12
-11/48
-2/3
k=9
-11/36
1/36
1/4
13/36
13/36
1/4
1/36
-11/36
-3/4
10
-23/60
-1/30
13/60
11/30
5/12
11/30
13/60
-1/30
-23/60
-5/6
S1(k,m) = -k / 12 (m+1) (m+1-k) / 2k
The white entries (for m<k) come directly from the decimation formula. Grey entries above the diagonal satisfy the recurrence (for p=1):
Sp(k,m+k) = Sp(k,m) (m+k)p
This establishes, by induction, the unified formula on the bottom line.
Of particular interest is the diagonal (m = k-1) wherethe sum of the decimated series is just k times the sum of the undecimated series (-1/12). Since the remaining terms are just the successive multiples of k in this case, we may divide by k and retrieve a series which differs from the original oneonly by being stretched with k-1 zero terms before each original term.
In our previous discussion of the constant series, we had already remarked that this type of padding always left the value (-1/2) of the sum unchanged.
To the uninitiated, that property may look more obvious than it actually is: Stretching a series this way often doesn't changethe sum, but it may...
To decimate the series of squares 1 + 4 + 9 + 16 + 25 + ... we use n* n2 zn-1 = f (z) with f (z) = (1+z)/(1-z)3 if z1 and f (1) = (-2) = 0.
S2
m = 0
m = 1
m = 2
m = 3
m = 4
m=5
m=6
m=7
m=8
m=9
k=1
0
k=2
0
0
k=3
-1/9
1/9
0
k=4
-1/4
0
1/4
0
k=5
-2/5
-1/5
1/5
2/5
0
k=6
-5/9
-4/9
0
4/9
5/9
0
k=7
-5/7
-5/7
-2/7
2/7
5/7
5/7
0
k=8
-7/8
-1
-5/8
0
5/8
1
7/8
0
k=9
-28/27
-35/27
-1
-10/27
10/27
1
35/27
28/27
0
10
-6/5
-8/5
-7/5
-4/5
0
4/5
7/5
8/5
6/5
0
11
-15/11
-21/11
-20/11
-14/11
-5/11
5/11
14/11
20/11
21/11
15/11
12
-55/36
-20/9
-9/4
-16/9
-35/36
0
35/36
16/9
9/4
20/9
13
-22/13
-33/13
-35/13
-30/13
-20/13
-7/13
7/13
20/13
30/13
35/13
14
-13/7
-20/7
-22/7
-20/7
-15/7
-8/7
0
8/7
15/7
20/7
S2(k,m) = -(m+1) (m+1-k/2) (m+1-k) / 3k
For cubes, n* n3 zn-1 = (1+4z+z2)/(1-z)4 if z1 or else (-3) = 1/120.
S3
m = 0
m = 1
m = 2
m = 3
m = 4
m = 5
m = 6
k = 1
1/120
k = 2
-7/120
1/15
k = 3
-13/120
-13/120
9/40
k = 4
-7/240
-7/15
-7/240
8/15
k = 5
29/120
-91/120
-91/120
29/120
25/24
k = 6
91/120
-13/15
-63/40
-13/15
91/120
9/5
k = 7
S3(k,m) = k3 / 120 (m+1)2(m+1-k)2/ 4k
On the diagonal m = k-1 the decimated sum is indeed k3/120 as expectedfrom the theorem heralded above, which is the subject of the next section.
Next, we consider the series of biquadrates (or fourth powers ). n* n4 zn-1 is 0 when z=1 and (1+11z+11z2+z3)/(1-z)5 when z1.
An infinite family of invariants, indexed by k, is formed by the sums of the k-decimations of a given series. Those form again a series whose invariants are second-order invariants of the original series, starting with the sum of the new series. This decimation procedure may becarried on with the new series, and so forth...
(2018-05-20) Harmonic analysis is a source of summable divergent series.
The Fourier expansion of the tangent function is a summable divergent series. So is the expansion of its derivative:
½ tan (x/2) = sin x sin 2x + ... + (-1)n+1 sin nx ... ¼ cos-2 (x/2) = cos x 2 cos 2x + ... + (-1)n+1 n cos nx ...
Don't even think about trying the latter with x = . However, with x = 0, that's the sum of a series we're alreadyfamiliar with. With x = /2, we get:
(2019-08-09) Definite differential properties of some divergent series.
Several methods exist to solve an ordinary differential equation as an asymptotic series about an hypothetical zero value of a nonzero parameter. The WKB method is one, but there are many others. In particular, Carl Bender has pointed out that the usual way to solve a perturbed Schrödinger equation expressesthe solution as a divergent asymptotic series.
Conversely, if we know a priori, the solutions corresponding to the values 0 and 1 of a parameter, such methods may yield a definite sum at point 1 for the ensuing asymptotic series about 0. Convergent or not.
Nicholas Mercator(né Niklaus Kauffman, 1620-1687) was the first to publish the series, in his Logarithmotechnia treatise (1668).
Isaac Newton (1643-1727) also discussed the series, which isnow called either the Mercator series or the Newton-Mercator series.
The Mercator series converges absolutely for |z| < 1. For z = -1, it's an alternating series that converges to -ln 2, as shown in 1650 byPietro Mengoli (1626-1686). For z = 1, we obtain the harmonic serieswhich was first shown to be divergent by Nicolas Oresme (c.1350).
(2018-07-01) Dealing with a multivalued analytic continuation.
f (1) is (aconstant we'll identifylater). For z1, we have essentially :
f (z) = - Log (1-z) / z
That's treacherous because Log is a multivalued complex function (definedmodulo 2i ) and wemust use a single analytic branch covering continuously the closed unit disk, with the possible exception of z=1. It turns out that all computer algebra systems (CAS) make the choice we needby implementing their Log function with a branch cut (or cliff) on the negative half of the real axis. For the sake of enlightenment, let's turn the auto-pilot off:
Consider a point z 1 on the unit circle: z = ei with
1-z
=
1 cos i sin
=
2 sin2 2 i sin cos
=
-2i sin ei
=
2 sin ei ()/2
In the above restricted range of , sin is positive. Therefore:
Log (1-z) = ln (2 sin /2) + i ()/2 + i 2n (for some integer n)
With n = 0, we get an analytic expression of f (z) valid onthe unit circlepunctured at z=1, consistent with f (0) = 1. The decimation formula yields:
1
k
exp (- 2i (m+1) q/k ) { ln (2 sin q/k) + i (q/k ½) }
Quick numerical verification, with arbitrary precision :
One easy way to validate numerically the result of the transfinite summation of a divergent series is to express that series as a sum ofother series which are either convergent series (whose sums are computed numerically with or without acceleration procedures) or divergent series of known sums. (Conveniently, the series of the k-th powers of the positive integershas zero sum when k is any positive even integer.)
To deal with the above case of the harmonic series, we'll use countable additivity to form a series with only three nonzero terms. Our starting point is provided by one of the formal series which define Bernoulli numbers (divided by n2, for good measure):
1
=
1
1
+
1
+ 0
n2
+ 0 ...
Bk nk-2
+ ...
n (en-1)
n2
2n
12
720
k!
n
1
=
2
1
+ 0 +
0
+ 0 ...
0
+ ...
n (en-1)
6
2
24
As the series on the left-hand-side converges nicely, itssum S is easily obtained numerically (0.6843...) to arbitrary precision, which yields:
= 2 / 3 1 / 12 2 S = 1.8378770664... = Log 2
Truthfully, the above verification is actually howthe transfinite sum of the harmonic function was first discovered (on 2018-07-12) using a 12-place decimal value (quickly obtained on a handheld calculator) which Simon Plouffe's Inverter identified immediately. A memorable magic moment !
At that time, I wasn't entirely sure that the postulate of countable additivity was legitimate, but that simple result gave me a great boost of confidence...
Arguably, is the correct value to assignto at the simple pole 1. The above goes against the popular guess that (1) ought to be defined as the Cauchy mean of about 1 (the constant term of theLaurent expansion):
If = n is the n-th element of an Appell sequence, this yields:
(2012-07-29)
If I tell you this, you will at once point out to me the lunatic asylum. Second letter of Ramanujanto G. H. Hardy (1913)
(2012-07-31) Anyp-adic series whose nth term is a multiple of n! converges !
The following series is clearlyconvergentin p-adic integers for any p :
1 + 1 + 2 + 6 + 24 + 120 + ... + n! + ...
That's because the result of the sum modulo pk is not influenced at all by the terms beyond a certain index m (namely, the least integer whose factorial is a multiple of pk ). This is also true if the radix (p) is not prime.
Thedecadic sum is ...4804323593105039052556442336528920420920314 The 2-adic sum is ...101110010110111111000011111011111101000011010
That series converges in p-adic integers for any radix p (prime or not) and the sum is not invertible for some of them, which may be perceived asits finite "factors". Those are the divisors of the following product:
Well before the more general notion of distributions was devised (in 1944, by my late teacherLaurent Schwartz) the Dutch mathematicianThomas Stieltjes considered measures as generalizedderivatives of functions of bounded variations of a real variable. Such functions are differences of two monotonous bounded functions;they need not be differentiable or continuous. (Stieltjes got hisdoctorate in Paris,underHermite andDarboux.)
The moment...
The poles are mocking up the cut...
(2012-07-30) (1955 & R.J. Schmidt 1941) The transform of a sequence has the same limit but better convergence.
Motivation :
In a convergent sequence of the form An = L + u vn we may extract the limit L from 3 consecutive terms, by eliminating u and v as follows:
An-1 = L + u vn-1
An = L + u vn
An+1 = L + u vn+1
So, v = ( An L )/ ( An-1 L ) = ( An+1 L )/ ( An L ) Therefore, ( An L ) 2 = ( An-1 L ) ( An+1 L ) which implies :
L = ( An-1 An+1 An2)/ ( An-1 + An+1 2 An )
Thus, the right-hand-side of that expression forms a sequence whose terms areexpected to be close to the limit of An even when An is not of the special form quoted above.
This motivates the following introduction of a new sequence Sn, which is defined for positive indices whenever the denominator doesn't vanish:
Shanks transform Sn of the sequence An
Sn =
An-1 An+1 An2
An-1 + An+1 2 An
We observe that Shanks' transformation commutes with translation :
Thus, wlg, we may focus on the analysis of sequences whose limit is zero. (The difference between a convergent sequence and its limit is of this type.)
An
Shanks Transform of An
vn
0
1 / n
1 / 2n
1 / np
~ 1 / (p+1)np
(-1)n / n
(-1)n+1 / [ 4n (n2- ½) ]
(-1)n / np
~ (-1)n+1 p / [ 4 np+2]
The table shows that the convergence of a sequence that alternates above and below its limitis greatly accelerated by Shanks' transformation (the distance to thelimit is essentially divided by the square of the index n). Shanks's transformation is thus highly recommended for alternating series.
No such luck when the sequence approaches its limit from one side only. The Shanks transform then offers only marginal improvement (by dividingthe distance to the limit by a constant factor, which is usually 2 or 3). In that case, the approach described in thenext sectionis preferred.
(2012-07-30) (1911 & Takebe 1722) Accelerating the convergence of An = L + k1 / n + k2 / n2 + ...
This (very common) pattern of convergence isthe case where the abovetransformation of Shanks has the poorest performance. By optimizing for this pattern, we'llprovide a convergence improvement in cases where the Shanks transformation does not deliver.
The method is similar, we eliminate 2 parameters between 3 equations:
An1 (n1)2 = L (n1)2 + k1 (n1) + k2
An n 2 = L n 2 + k1 n + k2
An+1 (n+1)2 = L (n+1)2 + k1 (n+1) + k2
Subtract twice the second equation from the sum of the other two:
An1 (n1)2 2An n 2 + An+1 (n+1)2 = 2 L
This motivates the definition of the (order 2) Richardson transformation:
Richardson transform Rn of the sequence An
Rn =
(n-1)2 An-1 2 n2 An + (n+1)2 An+1
2
Richardson's transform is a linear map that commutes with translation.
So, without loss of generality we can restrict the analysis of its performanceto convergent sequences whose limit is zero (consider such a sequenceas the difference between some other sequence and its limit, if you must).
An
Richardson Transform of An
1/ n
0
1/ n 2
0
1/ n 3
1/ n(n2-1)
1/ n 3
1/ n 4
(3n2-1)/n2(n2-1)2
3/ n 4
1/ n 5
(6n4-3n2+1)/n3(n2-1)3
6/ n 5
1/ n 6
(10n6-5n4+4n2-1)/n4(n2-1)4
10/ n 6
1/ n 7
(15n8-5n6+10n4-5n2+1)/n5(n2-1)5
15/ n 7
(-1)n/ n
2n (-1)n+1
(-1)n/ n 2
2 (-1)n+1
Thus, unlike theShanks transform, Richardson's transformation is absolutely catastrophic when applied to the partial sums of an alternating series. For a typical nonalternating series, it does a perfect job at the cancellationof the leading terms it's designed to handle and leaves the next order of magnitude virtually untouched. However, the bad influence of higher-order error terms is significantly amplified (possibly fatally so).
(2012-09-29) (Michon, 2012) Accelerating the convergence of An = L + k / (n-) + ...
The transformations presented in the previous section are somewhat unsatisfactorybecause they involve explicitly the particular indexation of the sequence (the value of n). Clearly, if we tune a convergence acceleration toa truncated expansion of the shape presented here, the index n won't be involvedbecause the presence of the parameter makes the optimal result invariant by translation of the index.
k.
Again, we eliminate 2 parameters between 3 equations:
An1 (n1) = L (n1) + k
An (n) = L (n) + k
An1 (n1) = L (n1) + k
Subtract twice the second equation from the sum of the other two:
( An1+ An+1 2 An)(n) + ( An+1 An1) = 0
Let's also subtract the second equation from the third:
( An+1 An)(n) + An+1 = L
Eliminating (n) between those two equations, we obtain:
This motivates the following definition of a new sequence Bn, which is valid for positive indices whenever the denominator doesn't vanish:
Transform Bn of the sequence An
Bn =
2 An1An+1 An(An1+ An+1)
An-1 + An+1 2 An
You may want to note that the Shanks transform of An is (An+Bn)/2.
As this new transform commutes with translation (the reader is encouraged to check that directly) we may study its performance, withoutloss of generality, for sequences whose limits are zero:
An
Bn =
Bn
1/ n
0
1/ n 2
-1/ (3n2-1)
-1/ 3n 2
1/ n 3
-3n/ (6n4-3n2+1)
-1/ 2n 3
1/ n 4
-(6n2+1)/(10n6-5n4+4n2-1)
-3/ 5n 4
(-1)n/ n
-2n (-1)n/ (2n2-1)
- (-1)n/ n
(-1)n/ n 2
-(2n2+1) (-1)n/ (2n4-n2-1)
- (-1)n/ n2
vn
-vn
Thus, the above transform is very effective when the leadingerror term is harmonic (1/n). For other types of convergence, the above table suggest usinga linear mix of A and B for best acceleration, as investigatednext.
(2012-09-30) Accelerating all typical analytical sequences.
Building onthe above,let's introduce a parameter u and define:
A'n = [ (1-u) An + (1+u) Bn] / 2
This way, the original sequence is obtained for u = -1, its Shanks transform for u = 0 and the sequence B of the previous section for u = 1.
A'n =
(1+u) An1An+1 u An(An1+ An+1) (1u) An2
An-1 + An+1 2 An
=
An1An+1 An2 u(An An1)(An+1 An)
An-1 + An+1 2 An
Or, equivalently:
Parametrized transformation of the sequence An
A'n = An + (1+u)
(An An1)(An+1 An)
(An An1)(An+1 An)
The invariance by translation of this parametrized transformallows us to study it only for sequences whose limit is zero (without loss of generality among convergent sequences). Of course, we'll seek the value of u which provides the best acceleration of convergence.
To use the example already analyzed, if An = L+k/n then
A'n = L + (1-u) k/n
As we already know, the best value of u is indeed +1 (which yields a constant sequence equal tothe limit L). Here are a few other cases:
Optimal values of u
An
u
A'n
1/ n
1
0
1/ n 2
1/2
-1/ 12n 4
1/ n 3
1/3
-1/ 6n 5
1/ n 4
1/4
-1/ 4n 6
1/ n 5
1/5
-1/ 3n 7
1/ n 6
1/6
-5/ 12n 8
1/ n p
1/p
(1-p)/ 12n p+2
(-1)n/ n
0
-(-1)n/ 4n 3
For the partial sums of alternating series, the Shanks transform (u=0) is optimal. Otherwise,we can typically build an optimized sequence as:
An , A'n , A''n , A'''n , A''''n , A'''''n , ...
For this, we use a special sequence of different parametersdetermined by the expected way the sequence approaches its limitasymptotically. Typically (but not always) the original sequenceapproaches its limit with a remainder roughly proportional to 1/n and one order of magnitude is gained with each iterationusing the sequence:
The computation is particularly easy to perform using a spreadsheet calculator. We illustrate this by the following computation to 6 decimal placesof thesum of the reciprocals of the squares,based on the first 7 terms in the series (9 terms are given to show that the last two are useless). Highlighted in blue are the Shanks transforms of the extreme diagonals.
(2) = 2/ 6 = 1.644934066848226436472415...
n
An
u0 = 1
u1 = 1/2
u2 = 1/3
u3 = 1/4
1
1.000000
1.644704
2
1.250000
1.650000
1.644921
3
1.361111
1.646825
1.644661
1.644934
4
1.423616
1.645833
1.644811
1.644934
5
1.463611
1.645429
1.644868
1.644934
1.644934
6
1.491389
1.645235
1.644895
1.644934
7
1.511797
1.645130
1.644909
1.644934
8
1.527422
1.645069
1.644931
9
1.539768
1.644909
Although many terms of the basic sequence would be easy to compute in thisdidactic example, the method is meant to handle situations where this is not the case. In theoretical physics (quantum field theory) and pure mathematics, we may only have a few terms available and onlya fuzzy understanding of the behavior of the sequence whose limit has to be guessedas accurately as possible.
Incidentally, with standard limited-precisionfloating-point arithmetic,the relevant computations presented above will be very poorly approximatedbecause we keep subtracting nearly-equal quantities. As a rule of thumb, about half the available precision is wasted. A 13-digit spreadsheet is barely good enough to reproduce the above6½-digit table. Extensions of it would be dominated by the glitches caused by limited precision.
Such pathological behavior is lessened by the approach described next.
(2012-10-02) The series counterpart of the parametrized transform for sequences.
If the sequence An is the partial sum of the series of term an , then we have an = An An1 (for n≥1) and theabove boils down to:
A'n = An + (1+u)
anan+1
anan+1
Subtracting from that value the counterpart for A'n-1 , we obtain:
a'n = an + (1+u)
anan+1
(1+u)
an1an
anan+1
an1an
(2012-09-30) An aborted attempt.
Let's target sequences of the form An = L + k1 / (n-) + k2 / (n-)2
For the purpose of the following computation, we get rid of indices byconsidering four consecutive terms in the sequence (A,B,C,D) and introducing the quantity x that differs from (n-) by some integer. We seek an expression of the limit L as a function of A,B,C,D by eliminating the three quantities x, k1 and k2 between the followingfour equations:
A (x+0) 2 = L (x+0) 2 + k1(x+0) + k2 [ 1] [ 1]
B (x+1) 2 = L (x+1) 2 + k1(x+1) + k2 [-3] [-2]
C (x+2) 2 = L (x+2) 2 + k1(x+2) + k2 [ 3] [ 1]
D (x+3) 2 = L (x+3) 2 + k1(x+3) + k2 [-1] [ 0]
The two columns of coefficients yield respectively these combinations:
( A - 3B + 3C - D ) x2 + ( -6B + 12C -6D ) x + ( -3B + 12C - 9D) = 0 ( A - 2B + C ) x2 + ( -4B + 4C ) x + ( -2B + 2C ) = 2L x
Eliminating x between those two quadratic equations yields:
A-3B+3C-D A-2B+C
-3B+12C-9D -2B+2C
=
A-3B+3C-D A-2B+C
-6B+12C-6D -2B+4C-2L
.
-6B+12C-6D -2B+4C-2L
-3B+12C-9D -2B+2C
Unfortunately, this is now a quadratic equation in L.
(2018-05-15) A forgotten method introduced by James Stirling in 1730.
(2017-06-24) Simplest rational function with prescribed truncatedTaylor expansion.
(2017-08-08) The bag of tricks physicists use to make sense out of divergent series.
(2019-11-21) Why divergent series may be unavoidable in quantum physics.
In the tamest type of infinite series, the absolutely convergent series, all the familiar propertie of finite sums are preserved, including commutativity and associativity. This comfortable fact may be useful for numerical applicationsbut it's simply roo restrictine for the way our quantum world seems to be constructed.
A certain historical order is essential to the way infinitely manyprobability amplitudes are combined, yet trajectories are not defined. There's no such thing as a partial history. Only the whole can be considered. The whole sum need not be the limit of partial sums.
Fractional derivatives are part of mathematical folklore. They were first defined in 1695 by Leibniz in a letterto Guillaume de l'Hospital. They apply in any context where fractional exponents are defined. They can be motivated by the following expression (easily established byinduction on k) for the continuous linear operator Dk defined as k repetions of theordinary derivation operator D = D1 when k is a positive integer (this expression is consistent with the convention that D0 is the identity operator) when it's applied to a monomial. By linearity, that definition extends to all polynomials and, by continuity,to all analytic functions.
Dk xm = [ m! / (m-k)! ] xm-k
Using theGamma function, we obtain an equivalent expression which remains valid when k isn't an integer.
Dk xm = [ (m+1) / (m-k+1) ] xm-k
That expression holds for almost all values of k, even complex ones (exceptions arise because is undefined for negative odd integers). In this, x is understood to be a positive real number (otherwise, raising x to a fractional power isproblematic). To remove that restriction, we may consider instead the compensated operator which isdefined in any power-associative algebra, including rings of matrices:
Jk xm = [ (m+1) / (m-k+1) ] xm
Jk doesn't change the degree of a polynomial, unless said degree is less than k.
(2018-06-23) 
(2018-05-15)