# Sum of the Reciprocal of the Difference of Two Squares

Is there a closed-form expression for the sum:

$$\sum \limits_{i=2}^{n} \frac{1}{i^2-1}$$

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What is a closed-form expression? +1 by the way. –  Sachin Kainth Oct 10 '12 at 23:22

Hint $$\frac{1}{i^2-1}=\frac{1}{2}\left(\frac{1}{i-1}-\frac{1}{i+1}\right)$$ Now use telescopy.

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Thanks a lot. It had actually occurred to me to express it as partial fractions, but after I did that I was stuck because I did not know about "telescopy", and the harmonic series (1/i) seemed nasty to deal with, so I assumed that this would be too. Clearly, I was wrong. Thanks again. –  Dara Oct 10 '12 at 22:47
@Dara You can accept this answer (by clicking checkbox under votes conunt) if you are satisfied with it. –  userNaN Oct 10 '12 at 22:54

Here is a closed form for the series

$$-\frac{1}{2}\,{\frac {1+2\,n}{ \left( n+1 \right) n}} + \frac{3}{4} \,.$$

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