Math 2.0

Sequences, series, summability Convergence tests Theorem

Abel's test

Builds on Convergent seriesMonotone sequenceBounded sequence

Statement

Let ∑an\sum a_n be a convergent real series. Let (bn)(b_n) be a monotone and bounded real sequence. Then the series ∑anbn\sum a_n b_n converges.

Remarks

The sequence (bn)(b_n) converges to some bb, by the monotone convergence theorem. Then ∑anbn=∑an(bn−b)+b∑an\sum a_n b_n = \sum a_n (b_n - b) + b \sum a_n. The first series converges by Dirichlet's test, since bn−bb_n - b is monotone with limit 00. The hypotheses on ∑an\sum a_n are stronger than in Dirichlet's test, and those on (bn)(b_n) are weaker.

History

Niels Henrik Abel proved the test in 1826.

Reading path

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

Show the path as a graph
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 sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionAbel's test, statement, no evidence levelAbel's 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. Bounded sequence definition
  4. Limit of a sequence definition
  5. Monotone sequence definition
  6. Series definition
  7. Convergent sequence definition
  8. Partial sum definition
  9. Convergent series definition
  10. Abel's test statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Abel%27s_test, Abel's test in real analysis
About this page
Tag0CN97GADE5
URIm2://0CN97GADE5
Kindstatement, theorem
WritingW1 stated
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0CN97GADE5.
UsesConvergent series, Monotone sequence, Bounded sequence
Same as

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge about out: the concepts that the statement is about (0 of 1)
  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
  • explanation L2 (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.