Convergence of infinite $ \sum (\frac{n}{n+1})^{n^2} $? I have being trying to solve this convergence but with no success. Using the ratio test I have reached here:
$$ 
a_n = \left( \frac{n}{n+1} \right)^{n^2}
$$
And
$$ 
\frac{1}{a_n} = \left( \frac{n+1}{n} \right)^{n^2}
$$
Also
$$ 
a_{n+1} = \left(\frac{n+1}{n+2}\right)^{{(n+1)}^2} = \left(\frac{n+1}{n+2}\right)^{n^2+2n+1} = \\ a_{n+1} = \left(\frac{n+1}{n+2}\right)^{n^2} \cdot \left(\frac{n+1}{n+2}\right)^{2n} \cdot \left(\frac{n+1}{n+2}\right)
$$
Therefore
$$
\lim \frac{a_{n+1}}{a_n} = \lim \frac{1}{a_n} \cdot a_{n+1}  = \\
\lim \left[ \left( \frac{n+1}{n} \right)^{n^2} \cdot \left(\frac{n+1}{n+2}\right)^{n^2} \cdot \left(\frac{n+1}{n+2}\right)^{2n} \cdot \left(\frac{n+1}{n+2}\right) \right]
$$
And I dont know how to go any further, I have tried many possibilities none of them reaching any result.

Instead of the ratio test, try the root test: 
$\displaystyle\lim_{n \to \infty}|a_n|^{1/n} = \lim_{n \to \infty}\left|\left(\dfrac{n}{n+1}\right)^{n^2}\right|^{1/n} = \lim_{n \to \infty}\left(\dfrac{n}{n+1}\right)^{n} = \lim_{n \to \infty}\left(1-\dfrac{1}{n+1}\right)^{n}$
Do you recongnize this limit?
