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.

Notice that $$\sqrt[n]n(2x+5)^n\sim_\infty (2x+5)^n$$
so by the ratio test we have
$$\left|\frac{(2x+5)^{n+1}}{(2x+5)^n}\right|=|2x+5|<1\iff -3<x<-2$$
hence the radius is $\frac12$ moreover for $x=-3$ or $x=-2$ the series is divergent since the general term of the series doesn't converge to $0$ in these cases. We conclude that $(-3,-2)$ is the interval of the convergence of the given series.
