When I recently read about the power rule for infinite sums, I thought that the sum $$\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}$$ would converge, as it is akin to a $p$-series, but with an (admittedly nonconstant) $p=1+\frac{1}{n} \ge 1$, and so is strictly less than the harmonic series. However, when I ask WolframAlpha for an evaluation, it claims that the sum diverges by the comparison test. I could accept this, but for the fact that I am not aware of a particular sum strictly lesser than my above sum that diverges. What might be a sum that fulfills that criterion?
2026-04-03 18:23:22.1775240602
Why A Sum Related To The Harmonic Series Diverges
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