Is there a closed-form expression for the sum:

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

  • $\begingroup$ What is a closed-form expression? +1 by the way. $\endgroup$ – 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.

  • $\begingroup$ 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. $\endgroup$ – Dara Oct 10 '12 at 22:47
  • $\begingroup$ @Dara You can accept this answer (by clicking checkbox under votes conunt) if you are satisfied with it. $\endgroup$ – Norbert Oct 10 '12 at 22:54

Here is a closed form for the series

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.