The Taylor series of $e^{\alpha x}\over 1+x^2$

Question

I'm interested in finding the Taylor expansion of $$e^{\alpha x}\over 1+x^2$$ in $x=0$ with an infinite radius of convergence. The method I tried, was expressing the Taylor expansion of $e^{\alpha x}$ and $1\over 1+x^2$ separately and combining them together, but the result was so terrific to pursue. Besides, the Taylor expansion of $1\over 1+x^2$ has a finite radius of convergence.

Is there any other way for calculating the derivatives of the function in $x=0$?

Answer 1

There is no such Taylor series, since the Taylor series of your function centered at $0$ has radius of convergence $1$ (because thats the distance from $0$ to the closest singularity).