Euler summation

Euler summation is a summability method for convergent and divergent series. Given a series Σa_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 particularly used to accelerate the convergence of alternating series and allows to evaluate divergent sums.

$_{E_{y}}\,\sum _{j=0}^{\infty }a_{j}:=\sum _{i=0}^{\infty }{\frac {1}{(1+y)^{i+1}}}\sum _{j=0}^{i}{i \choose j}y^{j+1}a_{j}.$

To justify the approach notice that for interchanged sum, Euler's summation reduces to the initial series, because

$y^{j+1}\sum _{i=j}^{\infty }{i \choose j}{\frac {1}{(1+y)^{i+1}}}=1$.

This method itself cannot be improved by iterated application, as

$_{E_{y_{1}}}{}_{E_{y_{2}}}\sum =\,_{E_{\frac {y_{1}y_{2}}{1+y_{1}+y_{2}}}}\sum$

Examples

• We have $\sum _{j=0}^{\infty }(-1)^{j}P_{k}(j)=\sum _{i=0}^{k}{\frac {1}{2^{i+1}}}\sum _{j=0}^{i}{i \choose j}(-1)^{j}P_{k}(j)$, if $P_{k}$ is a polynomial of degree k. Note that in this case Euler summation reduces an infinite series to a finite sum.

• The particular choice $P_{k}(j):=(j+1)^{k}$ provides an explicit representation of the Bernoulli numbers, since $\zeta (-k)=-{\frac {B_{k+1}}{k+1}}$. Indeed, applying Euler summation to the zeta function yields ${\frac {1}{1-2^{k+1}}}\sum _{i=0}^{k}{\frac {1}{2^{i+1}}}\sum _{j=0}^{i}{i \choose j}(-1)^{j}(j+1)^{k}$, which is polynomial for $k$ a positive integer; cf. Riemann zeta function.

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

See also

• Euler transform
• Borel summation
• Cesaro summation
• Lambert summation
• Abelian and tauberian theorems
• Van Wijngaarden transformation

References

• Korevaar, Jacob (2004). Tauberian Theory: A Century of Developments. Springer. ISBN 3-540-21058-X.
• Shawyer, Bruce and Bruce Watson (1994). Borel's Methods of Summability: Theory and Applications. Oxford UP. ISBN 0-19-853585-6.