Abel's Summation Formula

Theorem
Let $\sequence {a_n}_{n \in \N_{>0} }$ be a sequence in $\R$.

Let $f : \R_{\ge 1} \to \R$ be a continuously differentiable function.

Let $A : \R_{\ge 1} \to \R$ be defined as:
 * $\ds \map A x := \sum_{1 \mathop \le n \mathop \le x} a_n$

Then for all $x \in \R_{\ge 1}$:
 * $\ds \sum_{1 \mathop \le n \mathop \le x} a_n \map f n = \map A x \map f x - \int _1 ^x \map A u \map {f'} u \rd u$