# Thread: Find the sum of a series

1. ## Find the sum of a series

Hello there,

I'm a litttle bit stuck on a math question here

I need to find the Sum of the serie 1/((4*i^2)-1)
*made a mistake in the picture the i element is suppose to be a n sorry

I can't seem to understand how to even start

thank you guys

2. ## Re: Find the sum of a series

Let's look at partial sums: $S_k = \sum_{n=1}^k \dfrac{1}{4n^2-1}$. We have

$S_1 = \dfrac{1}{3}$

$S_2 = \dfrac{1}{3} + \dfrac{1}{15} = \dfrac{2}{5}$

$S_3 = \dfrac{2}{5} + \dfrac{1}{35} = \dfrac{3}{7}$

In general, the pattern appears to be $S_k = \dfrac{k}{2k+1}$. You can show inductively that this is the case.

So, $\sum_{n=1}^\infty \dfrac{1}{4n^2-1} = \lim_{k \to \infty} \dfrac{k}{2k+1} = \dfrac{1}{2}$.

3. ## Re: Find the sum of a series

Originally Posted by Memorystack
Hello there,

I'm a litttle bit stuck on a math question here

I need to find the Sum of the serie 1/((4*i^2)-1)
*made a mistake in the picture the i element is suppose to be a n sorry

I can't seem to understand how to even start

thank you guys
Notice that \displaystyle \begin{align*} \frac{1}{4n^2 - 1} = \frac{1}{(2n - 1)(2n + 1)} \end{align*}

Split it up into partial fractions and you should find that this is a telescopic series.