proof of radius of convergence

According to Cauchy’s root testMathworldPlanetmath a power seriesMathworldPlanetmath is absolutely convergent if

lim supk|ak(x-x0)k|k=|x-x0|lim supk|ak|k<1.

This is obviously true if

|x-x0|<1lim supk|ak|k=lim infk1|ak|k=.

In the same way we see that the series is divergent if

|x-x0|>lim infk1|ak|k,

which means that the right hand side is the radius of convergenceMathworldPlanetmath of the power series. Now from the ratio testMathworldPlanetmath we see that the power series is absolutely convergent if


Again this is true if


The series is divergent if


as follows from the ratio test in the same way. So we see that in this way too we can the radius of convergence.

Title proof of radius of convergence
Canonical name ProofOfRadiusOfConvergence
Date of creation 2013-03-22 13:21:50
Last modified on 2013-03-22 13:21:50
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 6
Author mathwizard (128)
Entry type Proof
Classification msc 40A30
Classification msc 30B10