calculus - Does $sum_{n=1}^infty left(frac{1}{n} - 1right)^n$ and $sum_{n=1}^infty left(frac{1}{n} - 1right)^{n^2}$ converge?

My task is this:

Determin whether $\sum_{n=1}^\infty \left(\frac{1}{n} - 1\right)^n$ and $\sum_{n=1}^\infty \left(\frac{1}{n} - 1\right)^{n^2}$ converge or diverge.

My thoughts:

For large $n$ one should expect $n^{-1} - 1 \to -1 \neq 0 $ raising that to $n$th power should yeald $\{-1,1\}$, again for big $n$, but I'm probably way off here and need some hints tips or better approach to this.

Thanks in advance!

Answer

Let $a_n = ( \frac{1}{n} - 1)^{n^2}$

Then $|a_n| = (1- \frac{1}{n})^{n^2} =\exp(n^2 \ln (1-\frac{1}{n})) \sim e^{-n+\frac{1}{2}}$

$ \sum e^{-n+\frac{1}{2}}$ converges so $ \sum |a_n|$ converges then $ \sum a_n$ also converges.

So the second sum does converge.

For the first sum, as already mentioned by H Potter, the general term does not converge to 0 so the sum does not converge.