Power Series Example

Question

In my notes for complex analysis, there is an example asking the Radius of convergence for the series $\sum_{n=0}^\infty (az)^{2n}$

In the answers it says that the coefficient for $z_n$, $a_n$ is equal to $a^n$ if n is even and 0 if n is odd

Can someone please explain why this is?

Thank you very much

Answer 1

Try to write out the first few terms: $$(az)^0+(az)^2+(az)^4+\dots$$ $$1+a^2z^2+a^4z^4+\dots$$ $$1+0z+a^2z^2+0z^3+a^4z^4+\dots$$

Answer 2

As $R^{-1}=\limsup\limits_n \,c_n^{1/n}$ by Hadamard's formula, we obtain at once that $$R=\frac1{|\mkern1mu a\mkern1mu|}.$$