I'm solving a visual problem, and sparing the boring details, I've found the sequence to be:

I need to solve for the sum of the series from 1 to infinity. Wolfram easily solves this as 2/3. However, I've poured over 2 calculus books and I've enlisted help but we can't solve it by hand.

Any help at simplifying this into a series which I can solve would be greatly appreciated.