Partial fractions seemed the most efficient route to take. However, I am having trouble at the end.
2026-03-28 22:55:26.1774738526
Having Trouble finding a simplified power series representation.
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What's your trouble at the end? $(-x)^n=(-1)^nx^n$, so you got $$\sum_{n=0}^{\infty}\left(2\cdot (-1)^n -1\right)x^n$$ Note the coefficients only depend on whether $n$ is even or odd.
In any event, it is possibly easier to do this as $$(1-3x)\cdot\frac{1}{1-x^2}=(1-3x)\left(1+x^2+x^4+\cdots\right)$$