TITLE: Convergence of a Series $\sum_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^{n^2}$- Which Test?
QUESTION [5 upvotes]: I tried root and ratio tests but it didn't work. Also, i can't use integral tests (and other "uncommon" ones) in this homework. 
(Prove that the series is convergent)
$$\sum_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^{n^2} = \frac{1}{2} + \left(\frac{2}{3}\right)^4 + \left(\frac{3}{4}\right)^9 + ...$$

REPLY [5 votes]: The Bernoulli Inequality says that for $n\ge1$,
$$
\left(1+\frac1n\right)^n\ge1+\frac{n}{n}=2
$$
Thus,
$$
\left(\frac{n}{n+1}\right)^{\large n^2}\le\left(\frac12\right)^n
$$
Thus, we can compare this to a geometric series.