Some special manipulations involving finite sums. How to solve this sum?
$\displaystyle{\sum_{k=1}^{n}}\frac{1}{4k^2 - 1}$
Some special manipulations involving finite sums. How to solve this sum?
$\displaystyle{\sum_{k=1}^{n}}\frac{1}{4k^2 - 1}$
Hint: Note that $\frac{1}{4k^2-1}=\frac{\frac{1}{2}}{2k-1}-\frac{\frac{1}{2}}{2k+1}.$
The following problem below is for an infinite series, but if you can solve it you may be able to solve your problem above.
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.
In the case that the sum converges, what is its sum?
Use (1) and (2) to show that $\sum_{k=0}^{\infty} \frac{1}{4k^2 - 1}$ converges and find its sum.