We have the power series:
$$ \sum_{n = 0}^{\infty}a_nz^n $$
The radius of convergence of that series is $R$.
Now I need to find the radius of convergence for:
$$ \sum_{n = 0}^{\infty}a_nc^nz^n, \quad c \in \mathbb C $$
The answers say that the new series converges for:
$$ |cz| < R $$
And then says that because it holds that:
$$ |cz| = |c||z| $$
We can conclude that the radius of convergence of the new series is: (for $|c| \neq 0$)
$$ R' = \frac{R}{|c|} $$
Why is the new radius not $R$, as we said that $|cz| < R$? Why do they go ahead and find $R'$?
$\sum a_nz^n$ converges if $|z|<R$ and diverges if $ |z|>R$.
thus
$\sum a_n(cz)^n$ converges if $|cz|<R $ or $|z|<\frac{R}{|c|}$
and diverges if $|cz|>R$ or $|z|>\frac{R}{|c|}$
hence, by definition, $$R'=\frac{R}{|c|}$$
$c $ is a real or a complex $ (\ne 0)$.