Math 2.0

Sequences, series, summability Series: basic notions Example

Harmonic series

Builds on SeriesDivergent series

Example

The harmonic series is the series ∑n=1∞1n=1+12+13+14+⋯ .\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots. Its terms tend to 00, but the series is a divergent series. Its partial sums are the harmonic numbers HN=∑n=1N1/nH_N = \sum_{n=1}^{N} 1/n, and they tend to +∞+\infty.

Remarks

The node harmonic series diverges gives a proof. It shows that the converse of the term test is false. The partial sums grow very slowly, like ln⁡N\ln N. The node harmonic numbers grow like log n makes this precise. The name comes from music. The wavelengths of the overtones of a string are 1/2,1/3,1/4,…1/2, 1/3, 1/4, \dots of the wavelength of the string.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionDivergent series, definition, no evidence levelDivergent seriesdefinitionHarmonic series, example, no evidence levelHarmonic seriesexample
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  1. Sequence definition
  2. Real sequence definition
  3. Limit of a sequence definition
  4. Series definition
  5. Convergent sequence definition
  6. Partial sum definition
  7. Convergent series definition
  8. Divergent series definition
  9. Harmonic series example

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Further reading

  1. Web https://en.wikipedia.org/wiki/Harmonic_series_(mathematics), Definition and divergence
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