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> Find if the following series converge: $$\displaystyle \sum\_{n=1}^{\infty} \left(\frac{n^2+1}{n^2+n+1}\right)^{n^2}$$
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What I did: 

$$a\_n=\left(\frac{n^2+1}{n^2+n+1}\right)^{n^2}$$

$$b\_n=\frac {1}{(n+1)^{n^2}}$$

$$\forall n\ge 1 : a\_n < b\_n \ \Rightarrow \ \displaystyle \sum\_{n=1}^{\infty}\left(\frac{n^2+1}{n^2+n+1}\right)^{n^2}<\displaystyle \sum\_{n=1}^{\infty}\frac {1}{(n+1)^{n^2}} \approx \displaystyle \sum\_{n=1}^{\infty} \frac 1 {n^2}$$

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> $\displaystyle \sum\_{n=1}^{\infty}b\_n$ converges like $\displaystyle \sum\_{n=1}^{\infty}\frac 1 {n^2}$ so $\displaystyle \sum\_{n=1}^{\infty}a\_n$ converges as well.
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