Subtracting two power series
I'm trying to find a power series representation for $$\frac{-x}{x^2 + 3x + 2}$$ I think I've done it correctly save the last step. I create partial fractions and have to subtract two different power series. I skip the more simple steps as I have verified those are correct. If more steps are needed please let me know
Get the partial fractions from our original equation
$$ \frac{1}{x+1} - \frac{2}{x+2} $$
Turn them into their respective power series
$$\sum_{n=0}^\infty (-1)^n(x)^n - \sum_{n=0}^\infty (-1)^n(\frac{x}{2})^n$$
This is where my math fails me, subtracting these just by subtracting their terms gives me
$$\sum_{n=0}^\infty (x^n - \frac{x^n}{2^n})$$
which equals the following.
$$\sum_{n=0}^\infty (\frac{2^nx^n - x^n}{2^n})$$
Now I know this is wrong, but I can't seem to make sense of my notes or any online resources to tell me where my mistake is. I'm very shaky on my power series so forgive me for simple mistakes. Thanks for reading and any help is appreciated!
$\endgroup$ 31 Answer
$\begingroup$You seem to have lost the $(-1)^n$. You should have $$\sum_{n=0}^\infty (-1)^n\frac{2^nx^n- x^n}{2^n}= \sum_{n=0}^\infty (-1)^n\frac{2^n- 1}{2^n}x^n \; .$$
$\endgroup$ 2
