boundedness of terms of power series

Theorem. If the set

$\{a_{0},\;a_{1}c,\;a_{2}c^{2},\;\ldots\}$

$\sum_{n=0}^{\infty}a_{n}z^{n}$

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

$|z|<|c|.$

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

$|a_{n}c^{n}|<M\quad\forall\,n\,=\,0,\,1,\,2,\,\ldots$

Thus one gets for the coefficients of the series the estimation

$|a_{n}|<\frac{M}{|c|^{n}}.$

If now $|z|<|c|$ , one has

$|a_{n}z^{n}|<M\left|\frac{z}{c}\right|^{n},$

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

Title	boundedness of terms of power series
Canonical name	BoundednessOfTermsOfPowerSeries
Date of creation	2013-03-22 18:50:44
Last modified on	2013-03-22 18:50:44
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	5
Author	pahio (2872)
Entry type	Theorem
Classification	msc 40A30
Classification	msc 30B10