I have the following series to test convergence \begin{align} S_{\infty}= \sum_{n=1}^{\infty} \dfrac{1}{n} \left( \dfrac{1}{n} \right)^{ \left( \dfrac{1}{n} \right) } < \sum_{n=1}^{\infty} \dfrac{1}{n} \ , \end{align} however it has several annoying properties that make it difficult to test convergence such as its derivatives create $\ln(n)$ terms, its terms are non-integrable in terms of a known function, and its finite sum is slightly below the harmonic series.
How does one test convergence for a tetration series slightly below the harmonic series?
Hint: $\displaystyle\lim_{n\to\infty}\sqrt[^n]n=1$.