proof of ratio test

Assume k<1. By definition N such that

i.e. eventually the series |an| becomes less than a convergentMathworldPlanetmathPlanetmath geometric seriesMathworldPlanetmath, therefore a shifted subsequenceMathworldPlanetmath of |an| converges by the comparison testMathworldPlanetmath. Note that a general sequenceMathworldPlanetmath bn converges iff a shifted subsequence of bn converges. Therefore, by the absolute convergence theorem, the series an converges.

Similarly for k>1 a shifted subsequence of |an| becomes greater than a geometric series tending to , and so also tends to . Therefore an diverges.

Title proof of ratio test
Canonical name ProofOfRatioTest
Date of creation 2013-03-22 12:24:46
Last modified on 2013-03-22 12:24:46
Owner vitriol (148)
Last modified by vitriol (148)
Numerical id 6
Author vitriol (148)
Entry type Proof
Classification msc 40A05
Classification msc 26A06