Exponential function as a sum

Question

I have an exercise that asks me to write $e^{2x}$ using a power series of $x+1$.

I know that $$e^{2x}=\sum_{n=0}^{\infty}\frac{(2x)^{n}}{n!}$$ Then, I tried something like this $$x=y+1\Rightarrow e^{2(y+1)}=\sum_{n=0}^{\infty}\frac{(2(y+1))^{n}}{n!}$$ But, I think that what the exercises asks is something like this: $$e^{2x}=\sum_{n=0}^{\infty}a_{n}(x+1)^{n}$$ isn't it?

Thanks ;)

Answer 1

Here are two facts you might find useful:

1. $$e^{2y+2} = e^2 e^{2y};$$

2. $$\frac{(2(y+1))^n}{n!} = \frac{2^n}{n!}(y+1)^n.$$