Find the general formula the taylor series of $e^{2x}$ and $xe^{2x}$

Question

$e^{2x}$ and $xe^{2x}$

I know that $$e^x=\sum_{n=0}^{\infty} \dfrac{x^{n}}{n!}$$.

So $$e^{2x} =\sum_{n=0}^{\infty} \dfrac{(2x)^{n}}{n!}$$

and $$xe^{2x}= \sum_{n=0}^{\infty} \dfrac{2^{n}x^{n+1}}{n!}$$

Is this correct, or it should be something else?

Answer 1

It is correct, except that those sums begin with $n=0$, not $n=1$.