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2 original proofs that harmonic series diverges

  1. All positive integers but \(1\) are divisble by at least one prime. The probability an integer is divisble by \(p\) is \(\frac{1}{p}\); the complement of that is \(1-\frac{1}{p}\) and so the product over all primes \(\displaystyle \prod_{p \ \text{prime}} \left( 1-\frac{1}{p} \right) = 0\). From the Euler product, we thus know \(\zeta(1)\) is divergent.

  2. Assume the harmonic series converges; then it converges absolutely, since all its terms are positive. This is obviously a contradiction, since we can tell it is conditionally convergent from the ratio/root test.

Note by Jake Lai
1 year, 6 months ago

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My proof for divergence requires no calculus.

\[\dfrac {1}{1} + \dfrac {1}{2} + \dfrac {1}{3} + \dfrac {1}{4} + \dfrac {1}{5} + \dfrac {1}{6} + \dfrac {1}{7} + \dfrac {1}{8} + \ldots > \dfrac {1}{1} + \dfrac {1}{2} + \dfrac {1}{4} + \dfrac {1}{4} + \dfrac {1}{8} + \dfrac {1}{8} + \dfrac {1}{8} + \dfrac {1}{8} + \ldots\]

\[\dfrac {1}{1} + \dfrac {1}{2} + \dfrac {1}{3} + \dfrac {1}{4} + \dfrac {1}{5} + \dfrac {1}{6} + \dfrac {1}{7} + \dfrac {1}{8} + \ldots > \dfrac {1}{1} + \dfrac {1}{2} + \left( \dfrac {1}{4} + \dfrac {1}{4} \right) + \left(\dfrac {1}{8} + \dfrac {1}{8} + \dfrac {1}{8} + \dfrac {1}{8} \right) + \ldots\]

\[\dfrac {1}{1} + \dfrac {1}{2} + \dfrac {1}{3} + \dfrac {1}{4} + \dfrac {1}{5} + \dfrac {1}{6} + \dfrac {1}{7} + \dfrac {1}{8} + \ldots > \dfrac {1}{1} + \dfrac {1}{2} + \dfrac {1}{2} + \dfrac {1}{2} + \ldots\]

\[\dfrac {1}{1} + \dfrac {1}{2} + \dfrac {1}{3} + \dfrac {1}{4} + \dfrac {1}{5} + \dfrac {1}{6} + \dfrac {1}{7} + \dfrac {1}{8} + \ldots > \infty\]

Hence proven. Sharky Kesa · 1 year, 6 months ago

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@Sharky Kesa Right! This is the classical proof of divergence of harmonic series using comparison. This proof is also featured in the Wikipedia page on Harmonic series. Hence, not an original proof. Prasun Biswas · 1 year, 6 months ago

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@Prasun Biswas Ye, but nonetheless, easy to remember. Sharky Kesa · 1 year, 6 months ago

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@Sharky Kesa Oresme reincarnate! Jake Lai · 1 year, 6 months ago

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@Jake Lai I'm surprised I know his name. Wasn't Nicole Oresme an astronomer, a mathematician and physicist? Sharky Kesa · 1 year, 6 months ago

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@Sharky Kesa Your proof is often attributed to Oresme. Jake Lai · 1 year, 6 months ago

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@Jake Lai Oh Sharky Kesa · 1 year, 6 months ago

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Your proof #1 just went over my head since I'm not quite familiar with Euler product and stuff yet. So, I won't comment on that.

Regarding your proof #2, how can you "tell it is conditionally convergent from the ratio/root test" ? Both ratio and root tests can only be used to test either absolute convergence or divergence. In other cases when the criteria for divergence or absolute convergence is not met, the test is inconclusive. When the tests are inconclusive, the sum can be convergent (absolute / conditional) or divergent. You cannot say for sure that the series must be conditionally convergent using ratio / root test in such inconclusive cases (as far as I know). I'm saying all this based on the Wikipedia articles on ratio / root tests.

Correct me if I'm wrong. Prasun Biswas · 1 year, 6 months ago

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