I have asked a couple of questions
Representing $\ln(x)$ as a power series centered at $2$ without computing any derivatives
Where I was trying to compute a Taylor series for a function while avoiding computing derivatives. I am hoping mathstack can clear up some of my confusion more generally.
As an example, I just tried to compute the Taylor series of $$ e^{x^2-x} $$ I am hoping to get some good guidelines (not necessarily rigorous, since "plugging in" isn't exactly a rigorous definition) for when you can just plug a function in to a Taylor expansion as follows $$ e^{x^2-x}=\sum_{k=0}^{\infty}\frac{(x^2-x)^k}{k!}=1+x^2+\frac{(x^2-x)}{2}+\frac{(x^2-x)^3}{6}+...... $$ I know it is correct to also think of the above expansion as the product of $e^{-x}$ and $e^{x^2}$, both of which are easy to expand and say convolve to get a specific degree coefficient, but is the above also fine?
My more general questions on problems of this type are:
1) Generally, what do I need to be aware of theoretically? Are there any theorems I am implicitly using in the above example, both when I plug in $x^2-x$ and when I take an infinite product? I think for the latter, I need both series to be absolutely convergent.
2) For convergence, you obviously inherit the restrictions on $x$ for convergence when you modify a series to compute another Taylor expansion, like when you integrate $\frac{1}{1+x^2}$ to compute $\arctan$'s expansion, you still have the restriction that $x^2<1$. But need those restrictions be as strict in the modified series as the original? In other words, am I losing some radius for $x$ by computing in this way? Can the restrictions sometimes get stricter depending on the operation you perform to modify the series?
Thanks