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.

Solution 1:

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.

Solution 2:

$$0\leq \sum_{n\geq 1}\frac{1}{\left(1+\frac{n}{n^2+1}\right)^{n^2}}\leq\sum_{n\geq 1}\frac{1}{\left(1+\frac{n^2}{n^2+1}\right)^n}=\sum_{n\geq 1}\frac{1}{\left(2-\frac{1}{n^2+1}\right)^n}\leq\sum_{n\geq 1}\frac{1}{(3/2)^n}=2.$$