Sum with radicals

Evaluate the sum

$$\mathcal{S}= \sqrt{1+\frac{1}{1^2}+ \frac{1}{2^2}}+ \sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\cdots + \sqrt{1+\frac{1}{2011^2}+\frac{1}{2012^2}}$$

Solution

We might begin from the well known established and celebrated identity:

$$\sqrt{1+\frac{1}{n^2}+\frac{1}{(n+1)^2}} = 1+\frac{1}{n} - \frac{1}{n+1}$$

Hence:

\begin{align*}
\mathcal{S} &=\sum_{k=1}^{2011} \left [ 1+ \frac{1}{k} - \frac{1}{k+1} \right ] \\
 &=2011 +1 - \frac{1}{2012} \\
 &=\frac{2012^2-1}{2012}
\end{align*}