I'm trying to understand something our teacher shoewd in class. The question was:
Find the taylor series of $f(x)=e^x$ around $x=0$ and show that it gives $e^x=e\,e^{x-1}$.
So, here's what she did:
$e + e(x-1) + \frac{e(x-1)^2}{2!}+\dots+\frac{e(x-1)^n}{n!}$ Is the series for $f(x)=e^x$ around $x=1$, which is also $e\left(1 + (x-1) + \frac{(x-1)^2}{2!}+\dots+\frac{(x-1)^n}{n!}\right) =e\,e^{x-1}$
Ok so I understand she just took the series for $f(x)=e^x$ arund $x=0$ and replaced $x$ with $x-1$. Can I just change the variable in any series I have? Or only if changing the variable does not change the derivative?
I am trying to understand the technique without getting into to much details, sorry if its a little vauge. Thanks!