Series Convergence / Divergence: $\sum \frac{3^{1/\sqrt{n}}-1}{n}$ Does the following series converge or diverge:
$$\sum_{n=1}^\infty \frac{3^{1/\sqrt{n}}-1}{n}$$
Many thanks!

Using MVT $3^x-1=3^{\epsilon}\ln{3}\cdot (x-0),\epsilon \in (0,x)$ or
$$0<\frac{3^{\frac{1}{\sqrt{n}}}-1}{n}=3^{\epsilon}\ln{3}\frac{1}{n\sqrt{n}}<3\ln{3}\frac{1}{n\sqrt{n}}$$
because $\epsilon \in (0,\frac{1}{\sqrt{n}})$ or $0<\epsilon<1$ and $f(x)=3^x$ is ascending. As a result:
$$\sum_{n=1}^{\infty}\frac{3^{\frac{1}{\sqrt{n}}}-1}{n}<3\ln{3}\sum_{n=1}^{\infty}\frac{1}{n^{1+\frac{1}{2}}}$$
which converges.
