Divergent series

Absolute convergence defines the sum of a sequence (or set) of numbers to be the limit of the net of all partial sums a_k1 + ... + a_kn, 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.

Cauchy's classical definition of the sum of a series a₀ + a₁ + ... defines the sum to be the limit of the sequence of partial sums a₀ + ... + a_n. This is the default definition of convergence of a sequence.

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

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

then the Cesàro sum C_k is defined by C_k(s) = N_(p^k)(s). Cesàro sums are Nørlund means if k ≥ 0, and hence are regular, linear, stable, and consistent. C₀ is ordinary summation, and C₁ is ordinary Cesàro summation. Cesàro sums have the property that if h > k, then C_h is stronger than C_k.

If λ_n = n, then we obtain the method of Abel summation. Here

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

where z = exp(−x). Then the limit of f(x) as x approaches 0 through positive reals is the limit of the power series for f(z) as z approaches 1 from below through positive reals, and the Abel sum A(s) is defined as

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

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

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

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

Then L(s), the Lindelöf sum (Volkov 2001), is the limit of f(x) as x goes to positive zero. The Lindelöf sum is a powerful method when applied to power series among other applications, summing power series in the Mittag-Leffler star.

If g(z) is analytic in a disk around zero, and hence has a Maclaurin series G(z) with a positive radius of convergence, then L(G(z)) = g(z) in the Mittag-Leffler star. Moreover, convergence to g(z) is uniform on compact subsets of the star.

If Σa_nxⁿ converges for small complex x and can be analytically continued along some path from x = 0 to the point x = 1, then the sum of the series can be defined to be the value at x = 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,[4] who observed that if ${\displaystyle 1\leq m<n}$ then

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

Evaluating at ${\displaystyle x=1}$, one gets

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

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

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

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

Euler summation is essentially an explicit form of analytic continuation. If a power series converges for small complex z 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 at q 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.

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

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

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

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

If the series

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

(for positive values of the a_n) converges for large real s and can be analytically continued along the real line to s = −1, then its value at s = −1 is called the zeta regularized sum of the series a₁ + a₂ + ... Zeta function regularization is nonlinear. In applications, the numbers a_i are sometimes the eigenvalues of a self-adjoint operator A with compact resolvent, and f(s) is then the trace of A^−s. For example, if A has eigenvalues 1, 2, 3, ... then f(s) is the Riemann zeta function, ζ(s), whose value at s = −1 is −1/12, assigning a value to the divergent series 1 + 2 + 3 + 4 + .... Other values of s can also be used to assign values for the divergent sums ζ(0) = 1 + 1 + 1 + ... = −1/2, ζ(−2) = 1 + 4 + 9 + ... = 0 and in general

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

where B_k is a Bernoulli number.[5]

In the special case when J(x) = e^x this gives one (weak) form of Borel summation.

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

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

where H is the second derivative of G and c(n) = e^−G(n), and a₀ + ... + a_h is to be interpreted as 0 when h < 0.

For example, if dμ = e^−x dx for positive x and 0 for negative x then μ_n = n!, and this gives one version of Borel summation, where the value of a sum is given by

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

There is a generalization of this depending on a variable α, called the (B′,α) sum, where the sum of a series a₀ + ... is defined to be

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

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

This summation method works by using an extension to the real numbers known as the hyperreal 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 ${\displaystyle \omega }$, which is used as a unit of infinity. Instead of summing to an arbitrary infinity (as is typically done with ${\displaystyle \infty }$), the BGN method sums to the specific hyperreal infinite value labeled ${\displaystyle \omega }$. Therefore, the summations are of the form

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

This allows the usage of standard formulas for finite series such as arithmetic progressions in an infinite context. For instance, using this method, the sum of the progression ${\displaystyle 1+2+3+\ldots }$ is ${\displaystyle {\frac {\omega ^{2}}{2}}+{\frac {\omega }{2}}}$, or, using just the most significant infinite hyperreal part, ${\displaystyle {\frac {\omega ^{2}}{2}}}$.[6]

Hardy (1949, chapter 11).

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 averages s₀ + s₁/2, s₁ + s₂/2, ..., and then taking the limit (Hardy 1949, p. 21).

The series a₁ + ... is called Ingham summable to s if

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

Albert Ingham showed that if δ is any positive number then (C,−δ) (Cesàro) summability implies Ingham summability, and Ingham summability implies (C,δ) summability Hardy (1949, Appendix II).

The series a₁ + ... is called Lambert summable to s if

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

If a series is (C,k) (Cesàro) summable for any k 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 Hardy (1949, Appendix II).

The series a₀ + ... is called Le Roy summable to s if

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

Hardy (1949, 4.11)

The series a₀ + ... is called Mittag-Leffler (M) summable to s if

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

Hardy (1949, 4.11)

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

The series a₁ + ... is called (R,k) (or Riemann) summable to s if

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

Hardy (1949, 4.17) The series a₁ + ... is called R₂ summable to s if

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

If λ_n form an increasing sequence of real numbers and

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

then the Riesz (R,λ,κ) sum of the series a₀ + ... is defined to be

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

The series a₁ + ... is called VP (or Vallée-Poussin) summable to s if

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

where ${\displaystyle \Gamma (x)}$ is the gamma function. Hardy (1949, 4.17).