Math 2.0

Sequences, series, summability Series: basic notions Proposition

Term test

Builds on Convergent seriesPartial sumLimit of a sum of sequences

Statement

Let ∑an\sum a_n be a convergent series of real numbers. Then an→0a_n \to 0 as n→∞n \to \infty. Equivalently, if (an)(a_n) does not converge to 00, then the series ∑an\sum a_n diverges.

Remarks

For the proof, let ss be the sum. Then an=sn−sn−1→s−s=0a_n = s_n - s_{n-1} \to s - s = 0. The converse is false. The terms of the harmonic series tend to 00, and the series diverges. For example, ∑(−1)n\sum (-1)^n and ∑n/(n+1)\sum n/(n+1) diverge by this test.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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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 sequencedefinitionEpsilon-N definition of the limit of a sequence, statement, no evidence levelEpsilon-N definition …statementPartial sum, definition, no evidence levelPartial sumdefinitionLimit of a sum of sequences, statement, no evidence levelLimit of a sum of seq…statementConvergent series, definition, no evidence levelConvergent seriesdefinitionTerm test, statement, no evidence levelTerm 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. Sequence definition
  2. Real sequence definition
  3. Limit of a sequence definition
  4. Series definition
  5. Convergent sequence definition
  6. Epsilon-N definition of the limit of a sequence statement
  7. Partial sum definition
  8. Limit of a sum of sequences statement
  9. Convergent series definition
  10. Term test statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Nth-term_test, Usage
  2. Web https://en.wikipedia.org/wiki/Nth-term_test, Proofs
About this page
Tag0VVY4H92G2
URIm2://0VVY4H92G2
Aliasesnth-term-test, divergence-test
Kindstatement, proposition
WritingW1 stated
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0VVY4H92G2.
UsesConvergent series, Partial sum, Limit of a sum of sequences
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