boundedness of terms of power series

Theorem.  If the set

$$\{a_0,a_{1}c,a_2{c}^2,\mathrm{\dots }\}$$

of the of a power series

$$\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n$$

at the point  $z=c$  is bounded (http://planetmath.org/BoundedInterval), then the power series converges, absolutely (http://planetmath.org/AbsoluteConvergence), for any value $z$ which satisfies

Proof.  By the assumption, there exists a positive number $M$ such that

Thus one gets for the coefficients of the series the estimation

If now  ,  one has

and since the geometric series ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\left|{\displaystyle \frac{z}{c}}\right|}^n$ is convergent, then also the real series ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}}|a_n{z}^n|$ converges.