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proof of divergence of harmonic series (by grouping terms)


The harmonic series can be shown to diverge by a simple argument involving grouping terms. Write

2Mn=11n=Mm=12mn=2m-1+11n.

Since 1/n1/N when nN, we have

2mn=2m-1+11n2mn=2m-1+12-m=(2m-2m-1)2-m=12

Hence,

2Mn=11nM2

so the series diverges in the limit M.

Title proof of divergence of harmonic series (by grouping terms)
Canonical name ProofOfDivergenceOfHarmonicSeriesbyGroupingTerms
Date of creation 2013-03-22 15:08:39
Last modified on 2013-03-22 15:08:39
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Proof
Classification msc 40A05