I have been set a few complex power series problems to solve and, while I feel fairly confident I know the general method of how to solve these, I feel that I am doing something wrong. If anyone could point out what I am doing wrong and guide me on how to get the correct answer I would appreciate it.
Find the radius of convergence for each of the following power series: $$1) \sum_{n=1}^{\infty }\frac{(z-1+i)^{n}}{2^{n}+n}$$
For this I simply attempted the ratio test, since it seemed the easiest:
$\lim_{n\to\infty} \left| \frac {a_{n+1}} {a_n} \right| = \lim_{n\to\infty} \left| \frac {(z-1+i)^{n+1}} {2^{n+1} + n+1} \frac{2^{n}+n}{(z-1+i)^{n}} \right| = \left|z-1+i \right| \lim_{n\to\infty} \left| \frac {a^{n+1} + (n+1)^2} {a^n + n^2} \right| \rightarrow |z-1+i|$ as $n \rightarrow \infty$ so $R=1$
$$2) \sum_{n=1}^{\infty }\frac{n!}{n^{n}}z^{n}$$
$=\sum_{n=1}^{\infty }n!(\frac{z^n}{n^{n}})^n$ so $\lim_{n\to\infty} \left| \frac {a_{n+1}} {a_n} \right| = \lim_{n\to\infty} \left| \frac {(n+1)!z^{n+1}} {(n+1)^{n+1}} \frac{n^{n}}{n!z^n} \right| = \lim_{n\to\infty} \left| \frac {n^nz} {(n+1)^n} \right| = |z| \lim_{n\to\infty} \left| \frac {n^n} {(n+1)^n} \right| \rightarrow |z|$ as $n \rightarrow \infty$ hence the radius of convergence is 1.
$$3)\sum_{n=1}^{\infty }\frac{n^{n}}{n!}z^n$$
$\lim_{n\to\infty} \left| \frac {a_{n+1}} {a_n} \right| = \lim_{n\to\infty} \left| \frac {(n+1)^{n+1}z^{n+1}} {(n+1)!} \frac{n!}{n^nz^n} \right| = \lim_{n\to\infty} \left| \frac {(n+1)^{n+1}z} {(n+1)n^n} \right| = |z| \lim_{n\to\infty} \left| \frac {(n+1)^n} {n^n} \right| \rightarrow |z|$ as $n \rightarrow \infty$ so the radius of convergence is 1.
I refuse to believe that all of these have the same answer, and I'm eager to know what I've done wrong.