proof of divergence of harmonic series (by splitting odd and even terms)

Suppose that the series ${\sum }_{n=1}^{\mathrm{\infty }}1/n$ 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:

$$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{2n}+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{2n-1}$$

Since ${\sum }_{n=1}^{\mathrm{\infty }}1/n=2{\sum }_{n=1}^{\mathrm{\infty }}1/(2n)$, we would conclude that

$$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{2n}=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{2n-1}.$$

But , hence , so we would also have

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)
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