Summation stuck under radical sign I am trying to evaluate the following sum, but I'm unable to solve it in any general way.
$$S=\sum_{k=1}^n\sqrt{1+\frac{1}{(k)^2}+\frac{1}{(k+1)^2} }$$
How can I do it?

The expression under the square root is $$\frac {k^4+2k^3+3k^2+2k+1} {k^2 (k+1)^2} = \frac {(1+k+k^2)^2} {k^2 (k+1)^2}$$ So your sum becomes $$\sum \limits _{k=1} ^n \frac {1+k+k^2} {k (k+1)} = \sum \limits _{k=1} ^n \left(1 + \frac 1 {k (k+1)} \right) = n + \sum \limits _{k=1} ^n \left(\frac 1 k - \frac 1 {k+1}\right) = n + 1 - \frac 1 {n+1}$$.
