What is a function with domain $(−∞, ∞)$ but that does not have a Taylor series centered at $0$? Why doesn’t this function have a Taylor series centered at $0$?
I tried $\ln(x)$ but the domain was wrong. Someone please help, I really don't understand Taylor series very well.
The canonical example is $f(x):=e^\frac{-1}{x^2}$, with $f(0)=0$. You can prove that $f^{(n)}(0)=0$ (the $n$th derivative) for every $n$ by induction. Thus, if $f$ had a Taylor series about $0$, we would have $$ f(x)=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n=0 $$ for every $x$ in some neighbourhood of $0$, which is absurd since $f(x)=0$ only for $x=0$.