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 for the left-hand side.