Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Some special manipulations involving finite sums. How to solve this sum?

$\displaystyle{\sum_{k=1}^{n}}\frac{1}{4k^2 - 1}$

share|improve this question
1  
Hint 1: the denominator can be factorised. Hint 2: Partial fractions. Hint 3: It telescopes. –  ShreevatsaR Oct 16 '11 at 17:09

2 Answers 2

The following problem below is for an infinite series, but if you can solve it you may be able to solve your problem above.

  1. Let $a_n := b_{n} - b_{n+1}$, for some other sequence $b_n$. Prove that the series $\sum_{n =0}^{\infty} a_n$ converges iff the sequence $b_n$ does.

  2. In the case that the sum converges, what is its sum?

  3. Use (1) and (2) to show that $\sum_{k=0}^{\infty} \frac{1}{4k^2 - 1}$ converges and find its sum.

share|improve this answer

Hint: Note that $$\frac{1}{4k^2-1}=\frac{\frac{1}{2}}{2k-1}-\frac{\frac{1}{2}}{2k+1}.$$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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