$1+\frac1{n^2}+\frac1{(n+1)^2}=\frac{n^4 + 2 n^3 + 3 n^2 + 2 n + 1}{(n^2+n)^2}=\left(1+\frac1{n^2+n}\right)^2$

It is a special case of the classical \[\frac{1}{(a-b)^2}+\frac{1}{(b-c)^2}+\frac{1}{(c-a)^2}=\left(\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}\right)^2.\]