Sequences and Series/Power series

{{proof|By Abelian partial summation, we have
 * $$\sum_{1 \le n \le x} a_n z^n = z^x A(x) - \ln(z) \int_1^x A(t) z^t dt$$

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
 * $$\sum_{1 \le n \le x} a_n \exp(wn) = \exp(wx) A(x) - w \int_1^x A(t) \exp(wt) dt$$.

We then put $$$$

{{BookCat}}