Sequences and Series/Power series

{{proof|By Abelian partial summation, we have

$\sum _{1\le n\le x}{a}_n{z}^n=z^{x}A(x)-\ln (z)\underset{1}{\overset{x}{\int }}A(t)z^{t}dt$

for $|z|<1$ and $x\ge 1$, where we denote as usual

$A(x):=\sum _{1\le n\le x}{a}_n$.

Substituting $z=\exp (w)$, we get

$\sum _{1\le n\le x}{a}_n\exp (wn)=\exp (wx)A(x)-w\underset{1}{\overset{x}{\int }}A(t)\exp (wt)dt$.

We then put ${\displaystyle }$