# Power Series expansion of $x\over(1+x-2x^2)$

I am unable to solve this specific problem.

The only "notable series expansion" I can use (and know) is $$\sum^{+\infty}_0 x^n =1\over(1-x)$$

I tried several things but none worked.

Writing $$x\over(1+x-2x^2)$$ as $$x * {1 \over(1-(-x+2x^2))}= x \sum(-x+2x^2)^n$$ did not help. Differentiation and integration also seem to not lead anywhere.

If it helps, the answer should be $$\sum {(1 - (-1)^n * 2^n) \over 3} * x ^n$$

Another thing is could not, and really tried, finding an online series representation calculator.

• First factorize the denominator, then break the fraction as a sum of 2 fractions, both of which could be expanded by using the notable series expansion you have mentioned. – xbh Apr 20 at 23:36
• Well, that solves it. I spent over 40min without thinking about it... Thank you – MTLaurentys Apr 20 at 23:41
• You are welcome. What I have said is exactly the same as mentioned by the answer below, i.e. do the partial fractions decomposition first. – xbh Apr 20 at 23:44

$$\frac{x}{(1+x-2x^2)}=\frac{1}{3 (1 - x)} - \frac{1}{3 (1 + 2 x)}$$