# boundedness of terms of power series

If the set

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

of the of a power series

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

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}|

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}|

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 BoundednessOfTermsOfPowerSeries 2013-03-22 18:50:44 2013-03-22 18:50:44 pahio (2872) pahio (2872) 5 pahio (2872) Theorem msc 40A30 msc 30B10