<s>[INST] How to find the sum of this : $\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+ \sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+.....$ How to find the sum of the following : 
$\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+ \sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+.....+\sqrt{1+\frac{1}{1999^2}+\frac{1}{2000^2}}$
Please suggest as getting no clue on this... thanks..
 [/INST] In addition to AWertheim's answer, in case you haven't studied telescopic series yet, this is how you can calculate $\displaystyle \sum_{n=1}^{1999} \frac{1}{n^{2}+n}$:
First you see that you can write:
 $ \displaystyle \frac{1}{n^{2}+n} =  \frac{1}{n} - \frac{1}{n+1} $ 
This technique is called partial fraction decomposition if you don't know it already.
Now see what happens:
$\displaystyle \sum_{n=1}^{1999} \frac{1}{n^{2}+n} = \displaystyle \sum_{n=1}^{1999}(\frac{1}{n} - \frac{1}{n+1}) = (1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4)  + \cdot \cdot \cdot +(1/1999 - 1/2000) = 1 + (-1/2+1/2) + (-1/3 +1/3) +\cdot \cdot \cdot + (-1/1999+1/1999)-1/2000= 1 - 1/2000 = \frac{1999}{2000}$
So, each term cancels the previous term, except the first and last terms. This kind of series is called telescopic series.
Also note that: $\displaystyle \sum_{n=1}^{1999} 1 = 1999$
So, your sum is:
$\displaystyle \sum_{n=1}^{1999} (1 + \frac{1}{n^{2}+n}) = \displaystyle \sum_{n=1}^{1999} 1 + \displaystyle \sum_{n=1}^{1999} \frac{1}{n^{2}+n} = 1999+ \frac{1999}{2000} = \frac{3999999}{2000} $
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