Convergence of $\sum_{n=1}^\infty (\frac{n}{n+1})^{n^2}$ Does the sum $\displaystyle \sum_{n=1}^\infty \left(\frac{n}{n+1}\right)^{n^2}$ converge?
I was given a hint to use the comparison test but fail to find a suitable series for comparison. I am not looking for a full solution but simply a convergent larger sequence for comparison.


 Given series is Convergent.

Hint : Use Cauchy's root test.
Let , $a_n=\left(\frac{n}{n+1}\right)^{n^2}$. Then , $a_n^{1/n}\to 1/e<1$.
