<s>[INST] determine whether the following series converge or diverge? determine whether the following series converge  or diverge ?
$a)$  $\sum_{n=1}^{\infty} (\frac {n}{n+1})^{n(n+1)}$
$b)$ $\sum_{n=1}^{\infty} (\frac {n^2 +1 }{n^2 +n +1})^{n^2}$
My attempts : for   $a)$ $(\frac {n}{n+1})^{n(n+1)}= (1- \frac{1}{n+1})^{n(n+1)}= e^{-n}$, now  $\sum_{n=1}^{\infty}\frac{1}{e^n}$ is  converge 
For  $b )$ $(\frac {n^2 +1 }{n^2 +n +1})^{n^2}=( 1-  \frac{n}{n^2+n+1})^{n. n}$...after  that  i can not able to procedd Further,,,,
Any hints/ solution will be appreciated
thanks in advance
 [/INST] By root test we have that
$$\sqrt[n]{a_n}=\left(\frac {n}{n+1}\right)^{n+1}=\left(1-\frac {1}{n+1}\right)^{n+1}\to \frac1e$$
$$\sqrt[n]{b_n}=\left(\frac {n^2 +1 }{n^2 +n +1}\right)^{n}=\left[\left(1-\frac {n}{n^2+n+1}\right)^{\frac{n^2+n+1}{n}}\right]^{\frac{n^2}{n^2+n+1}} \to \frac1e$$
therefore $\sum a_n$ and $\sum b_n$ both converge.
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