Math 2.0

Sequences, series, summability Series: basic notions Theorem

p-series test

Builds on Convergent seriesHarmonic series divergesCauchy condensation test

Statement

Let pp be a real number. The series ∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^p} converges if and only if p>1p > 1.

Remarks

For p=1p = 1 it is the harmonic series, which diverges. For p≤0p \le 0 the terms do not tend to 00. One proof uses the Cauchy condensation test. The condensed series is ∑2n⋅2−np=∑(21−p)n\sum 2^n \cdot 2^{-np} = \sum (2^{1-p})^n, a geometric series with ratio 21−p2^{1-p}. Another proof uses the integral test with f(x)=x−pf(x) = x^{-p}. For p>1p > 1 the sum is the value ζ(p)\zeta(p) of the Riemann zeta function. For example, ζ(2)=π2/6\zeta(2) = \pi^2/6, which is the Basel problem.

Reading path

To understand this page from the ground up, read these 18 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 sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSequence tending to infinity, definition, no evidence levelSequence tending to i…definitionSeries, definition, no evidence levelSeriesdefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementPartial sum, definition, no evidence levelPartial sumdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionDivergent series, definition, no evidence levelDivergent seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementCauchy condensation test, statement, no evidence levelCauchy condensation t…statementHarmonic series, example, no evidence levelHarmonic seriesexampleHarmonic series diverges, statement, no evidence levelHarmonic series diver…statementp-series test, statement, no evidence levelp-series teststatement
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. Bounded sequence definition
  5. Limit of a sequence definition
  6. Monotone sequence definition
  7. Sequence tending to infinity definition
  8. Series definition
  9. Convergent sequence definition
  10. Monotone convergence theorem for sequences statement
  11. Partial sum definition
  12. Convergent series definition
  13. Divergent series definition
  14. A series of nonnegative terms converges iff its partial sums are bounded statement
  15. Cauchy condensation test statement
  16. Harmonic series example
  17. Harmonic series diverges statement
  18. p-series test statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Harmonic_series_(mathematics), Riemann zeta function
About this page
Tag0ZNSSK8ZK1
URIm2://0ZNSSK8ZK1
Kindstatement, theorem
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0ZNSSK8ZK1.
UsesConvergent series, Harmonic series diverges, Cauchy condensation test

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