Math 2.0

Sequences, series, summability Definition

Divergent series

Builds on Convergent series

Definition

A series ∑an\sum a_n of real numbers is divergent if it is not a convergent series. Thus its sequence of partial sums has no limit in ℝ\mathbb{R}.

Remarks

The series 1+1+1+⋯1 + 1 + 1 + \cdots diverges, and its partial sums tend to +∞+\infty. The series 1−1+1−1+⋯1 - 1 + 1 - 1 + \cdots diverges, and its partial sums oscillate between 11 and 00. The harmonic series diverges, although its terms tend to 00. Summation methods, such as Cesàro summation, give values to some divergent series.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
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 sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionDivergent series, definition, no evidence levelDivergent seriesdefinition
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. Partial sum definition
  7. Convergent series definition
  8. Divergent series definition

Where this leads

14 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Divergent_series, lead section
About this page
Tag0NBRP6093V
URIm2://0NBRP6093V
Kinddefinition
WritingW1 stated
MSC40A05
DefinesDivergent series
UsesConvergent series

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge not_instance_of in: one non-example that fails one clause (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