Roots and Convergence of Power Series
We can express convergence or divergence of a power seriesin terms of the nth root of the nth term.
Letbe a power series.
If the sequenceis unbounded thenconverges only for
If the sequenceconverges to zero, thenconverges for all
If the sequenceis bounded andthenis the radius of convergence.
ChooseFor eachthere issuch that hencefor someIn particulardoes not converge to zero forhencediverges for
Supposeconverges to zero andDefine There issuch that forThus forso by the comparison test,converges for all
Suppose thatand consider anysuch that thenso there exist real numbersandsuch thatand forHence forthereforeconverges absolutely. To show thatis the radius of convergence, we must show thatdiverges forSupposethenhence there are infinitely manysuch thatso for infinitely manysodoes not converge to zero anddiverges.