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One of the methods for summing series of numbers and functions. A series
 
One of the methods for summing series of numbers and functions. A series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e0366001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\sum_{n=0}^\infty a_n\label{*}\tag{*}$$
  
is summable by means of the Euler summation method (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e0366002.png" />-summable) to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e0366003.png" /> if
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is summable by means of the Euler summation method ($(E,q)$-summable) to the sum $S$ if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e0366004.png" /></td> </tr></table>
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$$\lim_{n\to\infty}\frac{1}{(q+1)^{n+1}}\sum_{k=0}^n\binom nkq^{n-k}S_k=S,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e0366005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e0366006.png" />.
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where $q>-1$ and $S_k=\sum_{n=0}^k a_n$.
  
The method was first applied by L. Euler for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e0366007.png" /> to sum slowly-convergent or divergent series. Since the technique was later extended to arbitrary values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e0366008.png" /> by K. Knopp [[#References|[1]]], it is also known for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e0366009.png" /> as the Euler–Knopp summation method. This method is regular for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e03660010.png" /> (see [[Regular summation methods|Regular summation methods]]); if a series is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e03660011.png" />-summable, then it is also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e03660012.png" />-summable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e03660013.png" />, to the same sum (see [[Inclusion of summation methods|Inclusion of summation methods]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e03660014.png" /> the summability of the series (*) by the Euler summation method implies that the series is convergent. If the series is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e03660015.png" />-summable, then its terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e03660016.png" /> satisfy the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e03660017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e03660018.png" />. The Euler summation method can also be applied for analytic continuation beyond the disc of convergence of a function defined by means of a power series. Thus, the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e03660019.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e03660020.png" />-summable to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e03660021.png" /> in the disc with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e03660022.png" /> and of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036600/e03660023.png" />.
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The method was first applied by L. Euler for $q=1$ to sum slowly-convergent or divergent series. Since the technique was later extended to arbitrary values of $q$ by K. Knopp [[#References|[1]]], it is also known for arbitrary $q$ as the Euler–Knopp summation method. This method is regular for $q\geq0$ (see [[Regular summation methods|Regular summation methods]]); if a series is $(E,q)$-summable, then it is also $(E,q')$-summable, $q'>q>-1$, to the same sum (see [[Inclusion of summation methods|Inclusion of summation methods]]). For $q=0$ the summability of the series \eqref{*} by the Euler summation method implies that the series is convergent. If the series is $(E,q)$-summable, then its terms $a_n$ satisfy the condition $a_n=o((2q+1)^n)$, $q\geq0$. The Euler summation method can also be applied for analytic continuation beyond the disc of convergence of a function defined by means of a power series. Thus, the series $\sum_{n=0}^\infty z^n$ is $(E,q)$-summable to the sum $1/(1-z)$ in the disc with centre at $-q$ and of radius $q+1$.
  
 
====References====
 
====References====

Latest revision as of 17:36, 14 February 2020

One of the methods for summing series of numbers and functions. A series

$$\sum_{n=0}^\infty a_n\label{*}\tag{*}$$

is summable by means of the Euler summation method ($(E,q)$-summable) to the sum $S$ if

$$\lim_{n\to\infty}\frac{1}{(q+1)^{n+1}}\sum_{k=0}^n\binom nkq^{n-k}S_k=S,$$

where $q>-1$ and $S_k=\sum_{n=0}^k a_n$.

The method was first applied by L. Euler for $q=1$ to sum slowly-convergent or divergent series. Since the technique was later extended to arbitrary values of $q$ by K. Knopp [1], it is also known for arbitrary $q$ as the Euler–Knopp summation method. This method is regular for $q\geq0$ (see Regular summation methods); if a series is $(E,q)$-summable, then it is also $(E,q')$-summable, $q'>q>-1$, to the same sum (see Inclusion of summation methods). For $q=0$ the summability of the series \eqref{*} by the Euler summation method implies that the series is convergent. If the series is $(E,q)$-summable, then its terms $a_n$ satisfy the condition $a_n=o((2q+1)^n)$, $q\geq0$. The Euler summation method can also be applied for analytic continuation beyond the disc of convergence of a function defined by means of a power series. Thus, the series $\sum_{n=0}^\infty z^n$ is $(E,q)$-summable to the sum $1/(1-z)$ in the disc with centre at $-q$ and of radius $q+1$.

References

[1] K. Knopp, "Ueber das Eulersche Summierungsverfahren" Math. Z. , 15 (1922) pp. 226–253
[2] K. Knopp, "Ueber das Eulersche Summierungsverfahren II" Math. Z. , 18 (1923) pp. 125–156
[3] G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[4] S. Baron, "Introduction to theory of summation of series" , Tallin (1977) (In Russian)


Comments

References

[a1] K. Zeller, W. Beekmann, "Theorie der Limitierungsverfahren" , Springer (1970)
How to Cite This Entry:
Euler summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_summation_method&oldid=19089
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article