Radius of convergence of a complex serie Good evening, people.
I am trying to find the radius of convergente of the series $\sum\limits_{n=0}^{\infty} (4+(-1)^n )^n z^n$
I tried to use the theorem which states that $ r = \lim\limits_{n\rightarrow \infty} |\frac{a_{n}}{a_{n+1}}|$. However, I could not solve this limit.
So I did the following:
$$ \sum\limits_{n=0}^{\infty} (4+(-1)^n )^n z^n = \sum\limits_{k=0}^{\infty} 3^{2k+1}z^{2k+1} + \sum\limits_{k=0}^{\infty}5^{2k}z^{2k}$$
And said that the radius of convergence of the serie on the left side is the smallest radius between the two series on the right side.
Is this right? Is there an easy way to find the radius of convergence?
Thank you.

Yes, that is correct, except that in English the word "series" is either singular or plural, i.e. one writes "This series is..." or "These series are..."
There is a theorem that asserts that power series converge absolutely at every point in the interior of their circles of convergence (but may converge conditionally at a point on the boundary; however, that is of no concern here).
If convergence is absolute, then the series can be "rearranged" without altering the sum. In particular,
$$
\sum_{n\,=\,0}^\infty t_n = \sum_{n\,\in\,A} t_n + \sum_{n\,\in\,B} t_n
$$
if $A,B$ are disjoint sets whose union is $\{0,1,2,\ldots\}.$
If both of the series on the right converge absolutely, then so does the series on the left.
If the series on the left converges absolutely, then so do the two series on the right.
