Question

2. Ifanz" converges at z = z0, prove that anz" absolutely converges in the disk D(0, |zo|).

Answer 1

Suppose the power series _{$\sum_{n=0}^{\infty }a_nz_0^n$} converges for some z₀ with z₀ ̸= 0.

Then its terms converge to zero, so they are bounded and there exists M ≥ 0 such that

_{$|a_nz_0^n|\leq M$} for n = 0, 1, 2, . . . .

If |z| < |z₀|, then

_{$|{a_n}{z^n}|=|a_nz_0^n||\frac{z}{z_0}|\leq Mr^n$}where r < 1

Comparing the power series with the convergent geometric series _{$\sum_{n=0}^{\infty }Mr^n$} , we see that _{$\sum_{n=0}^{\infty }|{a_n}{z^n}|$} is absolutely convergent. Thus, if the power series converges for some z₀ , then it converges absolutely for every z with |z| < |z₀|.

So, it absolutely converges in the disk D(0,|z₀|).