Finite sum of the fractional type

Question

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

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

Answer 1

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.

Answer 2

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