$\begin{align*}
\frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2} + \frac{1}{(n+1)^2} &< \frac{n-1}{n} + \frac{1}{(n+1)^2} = \frac{(n-1)(n+1)^2+n}{n(n+1)^2} \\
&= \frac{(n^2-1)(n+1)+n}{n(n+1)^2} = \frac{n^3+n^2-1}{(n^2+n)(n+1)} \\
&< \frac{n^3+n^2}{(n^2+n)(n+1)} = \frac{n(n^2+n)}{(n^2+n)(n+1)}= \frac{(n+1)-1}{n+1}
\end{align*}$