Euler summation

Euler summation is a summability method for convergent and divergent series. Given a series Σ"a"_"n", if its Euler transform converges to a sum, then that sum is called the Euler sum of the original series.

Euler summation can be generalized into a family of methods denoted (E, "q"), where "q" ≥ 0. The (E, 0) sum is the usual (convergent) sum, while (E, 1) is the ordinary Euler sum. All of these methods are strictly weaker than Borel summation; for "q" > 0 they are incomparable with Abel summation.

Definition

Euler summation is particulary used to accelerate the convergence of alternating series and allows to evaluate divergent sums.:$\_\{E\_y\},\; sum\_\{j=0\}^infty\; a\_j\; :=\; sum\_\{i=0\}^infty\; frac\{1\}\{(1+y)^\{i+1\; sum\_\{j=0\}^i\; \{i\; choose\; j\}\; y^\{j+1\}\; a\_j\; .$

This method itself cannot be improved by iterated application, as :$\_\{E\_\{y\_1sum\; ,\; \_\{E\_\{y\_2sum\; =\; ,\; \_\{E\_\{frac\{y\_1\; y\_2\}\{1+y\_1+y\_2\}\; sum$

Examples

* $sum\_\{j=0\}^infty\; (-1)^j\; P\_k(j)\; =\; sum\_\{i=0\}^k\; frac\{1\}\{2^\{i+1\; sum\_\{j=0\}^i\; \{i\; choose\; j\}\; (-1)^j\; P\_k(j)$, if $P\_k$ is a polynomial of degree k.

* $sum\_\{j=0\}^infty\; z^j=\; sum\_\{i=0\}^infty\; frac\{1\}\{(1+y)^\{i+1\; sum\_\{j=0\}^i\; \{i\; choose\; j\}\; y^\{j+1\}\; z^j\; =\; frac\{y\}\{1+y\}\; sum\_\{i=0\}\; left(\; frac\{1+yz\}\{1+y\}\; ight)^i$. With appropriate choice of $y$ this series converges to $frac\{1\}\{1-z\}$.

ee also

* Euler transform
* Borel summation
* Cesaro summation

References

*cite book |last=Korevaar |first=Jacob |title=Tauberian Theory: A Century of Developments |publisher=Springer |year=2004 |id=ISBN 3-540-21058-X
*cite book |author=Shawyer, Bruce and Bruce Watson |title=Borel's Methods of Summability: Theory and Applications |publisher=Oxford UP |year=1994 |id=ISBN 0-19-853585-6