Fun problem: Prove that $$\sum_{k=1}^\infty\prod_{n=1}^k\frac{x^n}{1-x^n}=\sum_{k=1}^\infty\prod_{n=1}^k‌​\frac x{1-x^{2n}}$$ for all $|x|<1$.

You can obviously cancel the sum, as well as the product sign, as they are on both sides of the equation, but after that, I don't know how to proceed.

@flawr Uh, I don't think you can cancel the sum or product signs.

sum(1, 2, 3) == sum(2, 2, 2), but it's definitely not the case that 1 == 2 && 2 == 2 && 3 == 2.