Euler–Boole summation

Euler–Boole summation is a method for summing alternating series based on Euler's polynomials, which are defined by

${\displaystyle {\frac {2e^{xt}}{e^{t}+1}}=\sum _{n=0}^{\infty }E_{n}(x){\frac {t^{n}}{n!}}.}$

The concept is named after Leonhard Euler and George Boole.

The periodic Euler functions are

${\displaystyle {\widetilde {E}}_{n}(x+1)=-{\widetilde {E}}_{n}(x){\text{ and }}{\widetilde {E}}_{n}(x)=E_{n}(x){\text{ for }}0<x<1.}$

The Euler–Boole formula to sum alternating series is

${\displaystyle \sum _{j=a}^{n-1}(-1)^{j}f(j+h)={\frac {1}{2}}\sum _{k=0}^{m-1}{\frac {E_{k}(h)}{k!}}\left((-1)^{n-1}f^{(k)}(n)+(-1)^{a}f^{(k)}(a)\right)+{\frac {1}{2(m-1)!}}\int _{a}^{n}f^{(m)}(x){\widetilde {E}}_{m-1}(h-x)\,dx,}$

where ${\displaystyle a,m,n\in \mathbb {N} ,a<n,h\in [0,1]}$ and ${\displaystyle f^{(k)}}$ is the kth derivative.