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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.2012-10-10

2 Answers 2

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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.2012-10-10
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    @Dara You can accept this answer (by clicking checkbox under votes conunt) if you are satisfied with it.2012-10-10
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Here is a closed form for the series

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