proof of divergence of harmonic series (by splitting odd and even terms)
Suppose that the series ∑∞n=11/n converged. Since all the terms are positive, we could regroup them as we please, in particular, split the series into two series, that of even terms and that of odd terms:
∞∑n=11n=∞∑n=112n+∞∑n=112n-1 |
Since ∑∞n=11/n=2∑∞n=11/(2n), we would conclude that
∞∑n=112n=∞∑n=112n-1. |
But 2n-1<2n, hence 1/(2n)<1/(2n-1), so we would also have
∞∑n=112n<∞∑n=112n-1, |
which contradicts the previous conclusion. Thus, the assumption
that the
series converged is untenable.
Title | proof of divergence of harmonic series (by splitting odd and even terms) |
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Canonical name | ProofOfDivergenceOfHarmonicSeriesbySplittingOddAndEvenTerms |
Date of creation | 2013-03-22 17:38:26 |
Last modified on | 2013-03-22 17:38:26 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 4 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 40A05 |