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 evaluating 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 $\sum_{j=0}^\infty x^j P_k(j)= \sum_{i=0}^k \frac{x^i}{(1-x)^{i+1}}\sum_{j=0}^i {i \choose j} (-1)^{i-j} P_k(j)$, 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

• Borel summation
• Cesàro summation
• Lambert summation
• Perron's formula
• Abelian and tauberian theorems
• Abel–Plana formula
• Abel's summation formula
• 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. 
• Apostol, Tom M. (1974). Mathematical Analysis Second Edition. Addison Wesley Longman. ISBN 0-201-00288-4.