What is the power series of the given Maclaurin series

Question

The given series is: $$x+2x^2+x^3+2x^4+x^5+2x^6+....$$ Now I can see that the sum starts at $n=0$ and that is closely follows the common series $\frac{1}{1-x}$.

What I have so far is: $$\sum^\infty_{n=0}x^{n+1}$$ What I am having trouble with is the $2$ that alternates terms, I see that it is only present on the even exponent terms and that it must have an $(-1)^n$ term in the sum but I can't figure it out

Answer 1

Note that$$x^2+x^4+x^6+\cdots=\frac{x^2}{1-x^2}$$and that therefore the sum of your series is$$\frac x{1-x}+\frac{x^2}{1-x^2}=\frac{2x^2+x}{1-x^2}.$$

Answer 2

Note that it is $(x+2x^2)(1+x^2+x^4+x^6+\cdots)$