Function and Maclaurin series

Question

Function $f(x)=\frac{x^2+3\cdot\ e^x}{e^{2x}}$ need to be developed in Maclaurin series.

I can't find any rule to sum all fractions I've got...so any suggestion that helps?

Thanks

Answer 1

Hint: Do you know the expansion for $e^{-2x}$ and $e^{-x}$? Can you multiply a power series by $x^2$ and by $3$? If so, you have the tools. Just give the series for $x^2e^{-2x}+3e^{-x}$

Answer 2

Hint: $$ f(x)=\frac{x^2+3\cdot\ e^x}{e^{2x}} = f(x)={x^2e^{-2x}+3 \ e^{-x}} $$

$$ = x^2\sum_{k=0}^{\infty}\frac{(-2x)^k}{k!}+\dots .$$

Added:

$$ x^2\sum_{k=0}^{\infty}\frac{(-2x)^k}{k!}= \sum_{k=0}^{\infty} \frac{(-2)^k x^{k+2}}{k!}=\frac{1}{4}\sum_{k=2}^{\infty} \frac{(-2)^k x^{k}}{(k-2)!} .$$

Now, do the same with the other term.