First lets fix the format of the individual terms:
(1+n^{2}+(n+1)^{2})^{1/2}
=((n²(n+1)² + (n+1)² + n²)/(n²(n+1)²))
^{1/2}
=((n^4 + 2n^3 + 3n^2 + 2n + 1)/(n^2(n+1)^2))^(1/2)
=(n^2 + n + 1)/(n^2 + n)
=1 + 1/(n^2 + n)
So each term is 1 plus a small fraction. We can use partial fraction decomposition on this fraction so each term is now
=1 + 1/n  1/(n+1)
From this we see that in successive terms the 1/(n+1) term will cancel the 1/n term of the previous so that if we add up a bunch of successive terms we get
[number of terms] + 1/(first n)  1/(last n + 1)
The problem requests the sum of terms 1 to 2010 so the sum is
2010 + 1/1  1/2011
= 2011  1/2011
= 2010 + 2010/2011

Posted by Jer
on 20111225 17:23:09 