Summary: The paper gives a unified treatment of the summation of certain iterated series of the form $ \sum_{n=1}^{\infty}\sum_{m=1}^{\infty}a_{n+m},$ where $ (a_{n})_{n\in \mathbb{N}}$ is a sequence of real numbers. We prove that, under certain conditions, the double iterated series equals the difference of two single series.

40A05, 40B05

Bell numbers, iterated series, double alternating harmonic series, Stirling numbers, wolstenholme series