Power Series Example 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
Try to write out the first few terms:
$$(az)^0+(az)^2+(az)^4+\dots$$
$$1+a^2z^2+a^4z^4+\dots$$
$$1+0z+a^2z^2+0z^3+a^4z^4+\dots$$
As $R^{-1}=\limsup\limits_n \,c_n^{1/n}$ by Hadamard's formula, we obtain at once that
$$R=\frac1{|\mkern1mu a\mkern1mu|}.$$