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
2M∑n=11n=M∑m=12m∑n=2m-1+11n. |
Since 1/n≥1/N when n≤N, we have
2m∑n=2m-1+11n≥2m∑n=2m-1+12-m=(2m-2m-1)2-m=12 |
Hence,
2M∑n=11n≥M2 |
so the series diverges in the limit M→∞.
Title | proof of divergence of harmonic series (by grouping terms) |
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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 |