boundedness of terms of power series


Theorem.  If the set

{a0,a1c,a2c2,}

of the of a power seriesMathworldPlanetmath

n=0anzn

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

|z|<|c|.

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

|an|<M|c|n.

If now  |z|<|c|,  one has

|anzn|<M|zc|n,

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