Math 2.0

Sequences, series, summability Theorem

Harmonic series diverges

Builds on Harmonic seriesHarmonic numberSequence tending to infinity

Statement

The partial sums HN=∑n=1N1nH_N = \sum_{n=1}^{N} \frac{1}{n} of the harmonic series tend to +∞+\infty as N→∞N \to \infty. In particular, the harmonic series is a divergent series.

Remarks

The classical proof groups the terms in blocks from 2k−1+12^{k-1} + 1 to 2k2^k. Each block has 2k−12^{k-1} terms, each at least 1/2k1/2^k, so each block sums to at least 1/21/2. Thus H2k≥1+k/2H_{2^k} \ge 1 + k/2. A second proof uses the integral test: HN≥∫1N+1dx/x=ln⁡(N+1)H_N \ge \int_1^{N+1} dx/x = \ln(N + 1). Mathlib states it as the limit of ∑n<N1/(n+1)\sum_{n < N} 1/(n+1), in Real.tendsto_sum_range_one_div_nat_succ_atTop.

History

Nicole Oresme proved the divergence in the 14th century, with the grouping argument. Pietro Mengoli and Jacob Bernoulli gave other proofs in the 17th century.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Harmonic number, definition, no evidence levelHarmonic numberdefinitionSequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSequence tending to infinity, definition, no evidence levelSequence tending to i…definitionSeries, 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 seriesexampleHarmonic series diverges, statement, no evidence levelHarmonic series diver…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Harmonic number definition
  2. Sequence definition
  3. Real sequence definition
  4. Limit of a sequence definition
  5. Sequence tending to infinity definition
  6. Series definition
  7. Convergent sequence definition
  8. Partial sum definition
  9. Convergent series definition
  10. Divergent series definition
  11. Harmonic series example
  12. Harmonic series diverges statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Harmonic_series_(mathematics), Comparison test
About this page
Tag0E9M24KS9J
URIm2://0E9M24KS9J
Kindstatement, theorem
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0E9M24KS9J.
UsesHarmonic series, Harmonic number, Sequence tending to infinity

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