Euler's series transformation

The Euler transformation series generated from a convergent series of numbers to another set of numbers with identical row sum. The simple procedure was first applied to the Leibniz series by Nicolas Fatio and generalized to any series by Leonhard Euler . In some cases the transformed series will converge faster than the original series. This enables a better numerical calculation of the original series ( acceleration of convergence ). In some cases, this also opens up the possibility of evaluating the series sum using mathematical constants . In the case of the divergence of the original series, a series transformation can also provide a limitation method in that the transformed series converges to a value.

definition

The series and the transformed series are given by ${\displaystyle S}$ ${\displaystyle S^{\prime }}$

${\displaystyle {\begin{aligned}S&=\sum _{n=0}^{\infty }a_{n},\\S^{\prime }&={\tfrac {1}{2}}\sum _{n=0}^{\infty }2^{-n}\Delta ^{n}a_{0}={\tfrac {1}{2}}\sum _{n=0}^{\infty }2^{-n}\left(\sum _{k=0}^{n}{\binom {n}{k}}a_{k}\right).\end{aligned}}}$

Here the operator is defined by . In the case of an alternating series, produces differences in absolute values ​​of series terms. Except for a power of two and a sign, the terms of are the binomial transforms of . ${\displaystyle \Delta }$ ${\displaystyle \Delta a_{n}=a_{n}+a_{n+1}}$ ${\displaystyle \Delta }$ ${\displaystyle S^{\prime }}$ ${\displaystyle S}$

That the Euler transform gives the same series sum can be seen with the help of

${\displaystyle y^{k+1}\sum _{n=k}^{\infty }{n \choose k}{\frac {1}{(1+y)^{n+1}}}=1}$

verify ( y = 1 ).

Derivation

The idea of ​​Euler's series transformation (Nikolaus Fatio) consists in first creating a new series from the original series by combining successive series terms ${\displaystyle S=a_{0}/2+S_{1}}$

${\displaystyle S_{1}={\tfrac {1}{2}}\sum _{n=0}^{\infty }\left(a_{n}+a_{n+1}\right)={\tfrac {1}{2}}\sum _{n=0}^{\infty }\Delta a_{n}}$

to generate. For an alternating series with strictly monotonically decreasing absolute values, alternation is also used. Euler's series transformation is then obtained by repeated application of the method to the series generated in the previous step. ${\displaystyle S_{1}}$

Leonhard Euler reaches his goal in a different way. It defines (analogously) a function

${\displaystyle S\left(x\right)=\sum _{n=0}^{\infty }a_{n}x^{n+1},}$

sets , develops and sets , d. H. . ${\displaystyle x=-y/\left(1+y\right)}$ ${\displaystyle y}$ ${\displaystyle y=-{\tfrac {1}{2}}}$ ${\displaystyle x=1}$

Other series transformations

A comparison with other series transformations is possible if the partial sums of by the partial sums , by expressing, ${\displaystyle s_{N}^{\prime }}$ ${\displaystyle S'}$ ${\displaystyle s_{0}=a_{0}}$ ${\displaystyle s_{n}=s_{n-1}+a_{n}}$ ${\displaystyle S}$

${\displaystyle s_{N}^{\prime }=2^{-\left(N+1\right)}\sum _{n=0}^{N}{\binom {N+1}{n+1}}s_{n}.}$

The binomial coefficient approximates a Gaussian curve with mean value and standard deviation for large as a function of . The partial sum is therefore (asymptotically) a mean of partial sums of weighted with a Gaussian curve . ${\displaystyle N}$ ${\displaystyle n}$ ${\displaystyle \left(N-1\right)/2}$ ${\displaystyle {\sqrt {N+1}}/2}$ ${\displaystyle s_{N}^{\prime }}$ ${\displaystyle S}$

The Cesàro mean of a series, on the other hand, is the arithmetic mean of the partial sums . ${\displaystyle s_{n}}$

history

As early as 1730, James Stirling gave examples of series transformations in his Methodus differentialis .

Examples

• The series

${\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2^{k}}}={\tfrac {2}{3}}}$

provides the faster convergent series

${\displaystyle {\tfrac {1}{2}}\sum _{k=0}^{\infty }{\frac {1}{4^{k}}}={\tfrac {2}{3}}.}$

• However, Euler's series transformation does not provide a faster convergent series in all cases. For example

${\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{4^{k}}}={\tfrac {4}{5}}}$

the result is the slower convergent series

${\displaystyle {\tfrac {1}{2}}\sum _{k=0}^{\infty }\left({\tfrac {3}{8}}\right)^{k}={\tfrac {4}{5}}}$

• In the case of a divergent series, Euler's series transformation can represent a limitation method. For example

${\displaystyle \sum _{k=0}^{\infty }(-1)^{k}=1-1+1-\dots }$

the convergent series results

${\displaystyle {\tfrac {1}{2}}+0+0+0+\dots }$

One then says that the series can be E-limited .

• A less trivial application is the completely convergent series for the Dirichlet η function . ${\displaystyle \mathbb {C} }$

Further series transformations

In addition to Euler's series transformation, there are:

• Markoff series transformation
• Kummer's series transformation

literature

• James Stirling : Methodus Differentialis: sive Tractatus de Summatione et Interpolatione Serierum Infinitarum , G. Strahan, Londini (London) 1730 (Latin; Gallica )
• Konrad Knopp : Theory and Application of Infinite Series. Springer, Berlin, 1931
• Karl Strubecker: Introduction to higher mathematics. Oldenbourg, Munich et al. 1967, p. 226.
• L. Graham, DE Knuth, O. Patashnik: Concrete Mathematics. Addison-Wesley, 1994, ISBN 0201558025 , p. 199.
• Friedrich Lösch : A generalization of Euler's series transformation with applications of function theory. Springer, Berlin 1929, dissertation