boundedness of terms of power series

Theorem.  If the set


of the of a power seriesMathworldPlanetmath


at the point  z=c  is bounded (, then the power series convergesPlanetmathPlanetmath, absolutely (, for any value z which satisfies


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

|ancn|<Mn= 0, 1, 2,

Thus one gets for the coefficients of the series the estimation


If now  |z|<|c|,  one has


and since the geometric seriesMathworldPlanetmath n=0|zc|n is convergent, then also the real series n=0|anzn| 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