Finding if the given series converges or not: $\sum\limits_{n=1}^\infty \left(\frac{n^2+1}{n^2+n+1}\right)^{n^2}$ I want to find if the following series
$$\sum_{n=1}^\infty \left(\frac{n^2+1}{n^2+n+1}\right)^{n^2}$$
converges or not. 
I’ve tried root test so far, but reached 1 as the limit, which according to the related theorems, doesn’t yield any result. 

If $a_n$ is the $n-$th term then $\log a_n =-n^{2}\log (1+\frac n {n^{2}+1}) \leq -n^{2} \frac 1 2 (\frac n {n^{2}+1}) \leq -\frac n 4$ for $n$ sufficiently large. Comparing with $\sum e^{-n/4}$ we see that see that the series converges. 
