I'm trying to find complex power series expansion of $\frac{e^z}{1+z}$ centered at $z=0$ and its radius of convergence. Here is my attempt:
Since $e^z = \sum_{n=0}^\infty \frac{z^n}{n!}$, we can divide both terms by $1+z$ to get $e^z = \sum_{n=0}^\infty \frac{z^n}{(1+z)n!}$. Then I can get radius of convergence using the usual Cauchy-Hadamard formula.
Is this correct?
Thanks for any help!
Best Answer
In order to get the power series expansion we could multiply the series $e^z$ with $\frac{1}{1+z}$ using the Cauchy product
Comment:
In (1) we use the series expansion for the exponential function and the geometric series expansion
In (2) we multiply the series using the Cauchy product formula