$\sum_{n=1}^{\infty}(-1)^n\left(\frac {n+1}{n}\right)^{2n} $
This is not absolutely convergent because
$$\frac {1}{n}<\left(\frac {1}{n}\right)^{2n}<\left(1+\frac{1}{n}\right)^{2n}.$$ I am trying to show the conditional convergence, but the alternating series test is not working.
$$\lim_{n \to \infty}\left(1+\frac{1}{n}\right)^n\left(1+\frac{1}{n}\right)^n=e^2.$$
So, I was using the ratio test, but I don't know how to compute this: $$\left|\dfrac{\left(\frac{n+2}{n+1}\right)^{2n+2}}{\left(\frac{n+1}{n}\right)^{2n}}\right|.$$
I appreciate any hint.