I want to find the development of the function $ e^{x^2}$ and the set of convergence.
Considering that: $e^x= \sum_{k=0}^n \frac {x^k}{k!} + R_n(x,0)$
and substituting $x$ with $x^2$ I can write $e^{x^2}= \sum_{k=0}^n \frac {x^{2k}}{k!} + R_n(x,0)$
Now I have to prove that $ R_n(x,0)$ tends to $0$ to say that
$e^{x^2}= \sum_{k=0}^\infty \frac {x^{2k}}{k!}$
and for which $x$ it happens.
please help me