Find if $\sum\limits_{n=1}^{\infty} a^{1+\frac1{2}+\frac1{3}+\dots+\frac1{n}}$, a >0 converges or not. https://ift.tt/eA8V8J

Find if $\sum\limits_{n=1}^{\infty} a^{1+\frac1{2}+\frac1{3}+\dots+\frac1{n}}$, a >0 converges or not.

I used d'Alembert's criterion and I found $\lim_{n\to \infty} \frac{x_n}{x_{n+1}}=1$

Moving to Raabe–Duhamel's test, I found $\lim_{n\to \infty} n(\frac{x_n}{x_{n+1}} -1)$ = $\lim_{n\to \infty} n(\frac{1}{\sqrt[n+1]{a}} - 1) $

What can I do from there? Or is there a better way to find if $\sum\limits_{n=1}^{\infty} a^{1+\frac1{2}+\frac1{3}+\dots+\frac1{n}}$, a >0 converges or not?

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