Theorem (Abelian partial summation):
Let ( a n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }} be a sequence of complex numbers, and let f : [ 1 , ∞ ) → C {\displaystyle f:[1,\infty )\to \mathbb {C} } be differentiable on ( 1 , ∞ ) {\displaystyle (1,\infty )} . Finally define
Then for x ≥ 1 {\displaystyle x\geq 1} we have
Proof: If m = ⌊ x ⌋ {\displaystyle m=\lfloor x\rfloor } , we have
But
so that