How to find the convergence interval and radius of $\sum_{n=1}^{\infty}\sqrt[n]{n}(2x+5)^n$? Find the convergence interval and convergence radius of the sequence $$\sum_{n=1}^{\infty}\sqrt[n]{n}(2x+5)^n.$$
I tried ratio test: $$\lim_{n\to \infty}\left|\frac {\sqrt[n+1]{n+1}(2x+5)^{n+1}}{\sqrt[n]{n}(2x+5)^n}\right|=|(2x+5)|\lim_{n\to \infty}\left|\frac {\sqrt[n+1]{n+1}}{\sqrt[n]{n}}\right|.$$
But couldn't conclude.

You're almost right, just log the expressions:
$$
\frac{(n+1)^{\frac{1}{n+1}}}{n^{\frac{1}{n}}} \rightarrow \frac{\log(n+1)}{n+1} - \frac{\log n}{n} \to_n0-0=0
$$
hence the limit of the ration is $e^0=1$. So allyou need to do is check $|2x+5|<1$.
