Euler summation method

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)

References

[a1]	K. Zeller, W. Beekmann, "Theorie der Limitierungsverfahren" , Springer (1970)