Complex Analysis - Series Analytic Regions

[WannaBe22]

Homework Statement
Find the analytic regions of the next functions:

A. $$f(z)=\sum_{n=1}^{\infty} [ \frac{z(z+n)}{n}]^n$$
B. $$f(z)=\sum_{n=1}^{\infty} 2^{-n^2 z} \cdot n^n$$

Homework Equations

The Attempt at a Solution

In the first one: I've tried writing : $$f(z)= \sum \frac{z^{2n}}{n^n} + \sum z^n$$
and the second element in the sum converges iff |z|<1... Is it enough?

About B: We can write this series as: $$\sum [ \frac{n}{2^{nz}}]^n$$ ... But I don't think it helps us...

Hope you'll be able to help me


TNX!

 

[Eynstone]

Homework Statement
Find the analytic regions of the next functions:

A. $$f(z)=\sum_{n=1}^{\infty} [ \frac{z(z+n)}{n}]^n$$
B. $$f(z)=\sum_{n=1}^{\infty} 2^{-n^2 z} \cdot n^n$$

The first series ultimately grows like z^n exp(z) & hence converges for |z|<1.
The second problem can be tackled similarly.