Math 2.0

Sequences, series, summability Proposition

A series of nonnegative terms converges iff its partial sums are bounded

Builds on Convergent seriesPartial sumMonotone convergence theorem for sequences

Statement

Let (an)(a_n) be a real sequence with an≥0a_n \ge 0 for all nn. Then the series ∑an\sum a_n converges if and only if its partial sums sN=∑n=0Nans_N = \sum_{n=0}^{N} a_n are bounded above. In this case, ∑n=0∞an=sup⁡NsN\sum_{n=0}^{\infty} a_n = \sup_N s_N.

Remarks

The partial sums increase, since sN+1−sN=aN+1≥0s_{N+1} - s_N = a_{N+1} \ge 0. The result is then the monotone convergence theorem. If the partial sums are not bounded, they tend to +∞+\infty, and one writes ∑an=+∞\sum a_n = +\infty. The comparison test, the integral test and the Cauchy condensation test all rest on this fact.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Sequence, 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 sequencedefinitionSeries, 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 seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…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. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Limit of a sequence definition
  5. Monotone sequence definition
  6. Series definition
  7. Convergent sequence definition
  8. Monotone convergence theorem for sequences statement
  9. Partial sum definition
  10. Convergent series definition
  11. A series of nonnegative terms converges iff its partial sums are bounded statement

Where this leads

123 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Monotone_convergence_theorem, Convergence of a monotone series
About this page
Tag0MX1HZBBXJ
URIm2://0MX1HZBBXJ
Kindstatement, proposition
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0MX1HZBBXJ.
UsesConvergent series, Partial sum, Monotone convergence theorem for sequences

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