Study the convergence of the series $\sum_{n=1}^\infty \left( \sqrt[17]{5+ \frac1n} - \sqrt[17]{5}\right)^{a}q^n$ for $a>0$ and $q\in\mathbb R$

Question

I have a problem with this task, because I think the most important is idea to do convergence of the series $$ \sum_{n=1}^\infty\left( \sqrt[17]{5+ \frac{1}{n} } - \sqrt[17]{5}\right)^{\!a}, $$ but it is difficult for me because it is power $a$ and for $a=1$.

I can use claim about three series, but in this case I completely don't knew what can I do.

Answer 1

Hint. First show that $$ \lim_{n\to\infty\,}n\left(\sqrt[17]{5+\frac{1}{n}}-\sqrt[17]{5}\right)=\frac{\sqrt[17]{5}}{85}, $$ and hence $$ \left(\sqrt[17]{5+\frac{1}{n}}-\sqrt[17]{5}\right)^a\approx \left(\frac{\sqrt[17]{5}}{85}\right)^an^{-a} $$ Thus, we have convergence of the series if $|q|<1$, or $q=1$ and $a>1$, or $q=-1$ and $a>0$.