I am having trouble characterizing the types of alternating with-differing-period harmonic series which converge. For example, $\sum_{n\in \mathbb{N}} (-1)^n\frac{1}{n}$ converges, but what about $1+\frac{1}{2} - \frac{1}{3} + \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7} + \frac{1}{8} - \frac{1}{9} \dots$ ($2$ positive terms and $1$ negative term). Can we characterize how far the gaps can be between negative terms?
2026-02-23 00:21:13.1771806073
Modified alternating harmonic series
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