Assume that has a power-series representation in terms of powers of , verify that it has the form

\[ \frac{x}{(1-x)(1-x^2)} = \frac{1}{2} \sum_{n=1}^{\infty} \left( n + \frac{1 – (-1)^n}{2} \right)x^n

valid for all real such that .

First, we rewrite the given expression in a form we can deal with,

Then, we use the geometric series formula, and we call from a previous exercise (Section 11.13, Exercise #3) that , so we have

This valid for all real since that was where the geometric series formulas we used were valid.