Math 2.0

Sequences, series, summability Convergence tests Theorem

Alternating series test

Builds on Alternating seriesMonotone sequenceMonotone convergence theorem for sequences

Statement

Let (bn)(b_n) be a decreasing real sequence with bn→0b_n \to 0 as n→∞n \to \infty. Then the alternating series ∑n=0∞(−1)nbn\sum_{n=0}^{\infty} (-1)^n b_n converges. Moreover, its sum SS and its partial sums sN=∑n=0N(−1)nbns_N = \sum_{n=0}^{N} (-1)^n b_n satisfy ∣S−sN∣≤bN+1|S - s_N| \le b_{N+1} for each NN.

Remarks

The even partial sums s0,s2,s4,…s_0, s_2, s_4, \dots decrease, and the odd partial sums s1,s3,…s_1, s_3, \dots increase. Each odd partial sum is below each even one, so both sequences converge by the monotone convergence theorem. Their limits agree, since s2k−s2k+1=b2k+1→0s_{2k} - s_{2k+1} = b_{2k+1} \to 0. The hypothesis that (bn)(b_n) decreases is needed. Without it, an alternating series with terms tending to 00 can diverge. Mathlib states the convergence for an antitone sequence, in Antitone.tendsto_alternating_series_of_tendsto_zero.

History

Leibniz discussed the criterion in an unpublished work of 1676. For this reason it is also called the Leibniz test.

Reading path

To understand this page from the ground up, read these 9 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 levelSeriesdefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementAlternating series, definition, no evidence levelAlternating seriesdefinitionAlternating series test, statement, no evidence levelAlternating series te…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. Monotone convergence theorem for sequences statement
  8. Alternating series definition
  9. Alternating series test statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Alternating_series_test, Formal statement
About this page
Tag0PK8QH1W1V
URIm2://0PK8QH1W1V
Aliasesleibniz-test
Kindstatement, theorem
WritingW1 stated
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0PK8QH1W1V.
UsesAlternating series, Monotone sequence, Monotone convergence theorem for sequences
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

Used by