I wasn't able to find if they converges or not :

1)$\displaystyle \sum_{n=1}^{+\infty} \left( \frac{n}{2n+1}\right) ^n$ My intuition tells me it converges.

And 2)$\displaystyle \sum_{n=1}^{+\infty} (\sqrt[n]{n}-1)^n$. This one is going to make me crazy. I'm not sure at all if it diverges or not.

For both I've tried the root test, integral test and for the first the ratio test but in each case I failed to get a conclusion. I need a little help (maybe the test to do, and any trick I might not see) but not the full detailed answer, please.