Why doesn't $\sum_{n=1}^\infty \frac{1}{n^{1+\frac{1}{n}}}$ converge?

Question

$\sum_{n=1}^\infty \frac{1}{n^{1+\frac{1}{n}}} = \infty$. Is there a comparison that works well to prove this?

Answer 1

Hint: Show that $n^{1/n}$ is bounded above, so that $\frac{1}{n^{1+1/n}}>\frac{C}{n}$ for some constant $C$.

Answer 2

Alternatively, you can use the limit comparison test. Since

$$\lim_{n\to \infty} n\cdot \frac{1}{n^{1 + 1/n}} = \lim_{n\to \infty} \frac{1}{n^{1/n}} = 1$$

and $\sum_{n = 1}^\infty \frac{1}{n}$ diverges, your series diverges by limit comparison.