# Thread: Two Series & Equality

1. ## Two Series & Equality

I have no idea where to start with this one. I tried finding the limits of each, finding the partial sums, I just don't really know how to prove it

Show that

$\frac{\(-1)^(n+1)}{n}$ = $\frac{\n(2n-1)}{1}$

any help would be appreciated

2. Originally Posted by starfish
I have no idea where to start with this one. I tried finding the limits of each, finding the partial sums, I just don't really know how to prove it

Show that

$\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}=\sum_{n=1}^\infty\frac{1}{2n(2n-1)}$

any help would be appreciated
Did I guess right in correcting your (messy) formula? (you can click on it to see what I wrote)

If so, you should cut the left-hand side sum into two parts: the even (the numerator is -1) and the odd terms (it is +1), and then put both sums together.

Another way (this is the same in fact) is summing the terms two by two: write $\sum_{n=1}^\infty a_n=(a_1+a_2)+(a_3+a_4)+\cdots=\sum_{p=1}^\infty( a_{2p-1}+a_{2p})$ for the left-hand side.