# 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) |
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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 |