$f(z) = \sum_{n=0}^{\infty} c_n z^n$ has radius of convergence R. Show $\sum_{n=0}^{\infty} \overline{c_n} z^n$ has same radius of convergence

Question

suppose $f(z) = \sum_{n=0}^{\infty} c_n z^n$ is a power series with finite radius of convergence $R > 0$. Show $\sum_{n=0}^{\infty} \overline{c_n} z^n$ has the same radius of convergence, R

I'm super stuck on this and I haven't learnt Hadamard's Theorem so please don't suggest it!

Answer 1

The series $\sum_{n=0}^\infty c_nz^n$ converges if and only if the series $\sum_{n=0}^\infty\overline{c_n}\,\overline{z}^n$ converges. Besides, $\bigl|\overline z\bigr|=|z|$.