Sequences and Series/Series and integration

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Theorem (Abelian partial summation):

Let ${\displaystyle ({�}_{�}{)}_{�\in \mathbb{�}}}$ be a sequence of complex numbers, and let ${\displaystyle �:[1,\mathrm{\infty })\to \mathbb{�}}$ be differentiable on ${\displaystyle (1,\mathrm{\infty })}$. Finally define

${\displaystyle �(�):=\sum _{1\le �\le �}{�}_{�}}$.

Then for ${\displaystyle �\ge 1}$ we have

${\displaystyle \sum _{1\le �\le �}{�}_{�}�(�)=�(�)�(�)-{\int }_1^{�}�(�){�}^{\prime }(�)��}$.

Proof: If ${\displaystyle �=\lfloor �\rfloor }$, we have

But

so that

${\displaystyle {\int }_1^{�}�(�){�}^{\prime }(�)��=�(�)�(�)-\sum _{�=1}^{�}{�}_{�}�(�)}$.