Math 2.0

definition

Convergent series

Definition

Let ∑an\sum a_n be a series of real numbers with partial sums sN=∑n=0Nans_N = \sum_{n=0}^{N} a_n. The series is convergent if the sequence (sN)(s_N) is a convergent sequence.

Remarks

The series ∑1/2n\sum 1/2^n converges, since its partial sums 2−1/2N2 - 1/2^N tend to 22. The limit of the partial sums is the sum of the series. A series that does not converge is a divergent series. Changing finitely many terms does not change whether a series converges. It can change the sum.

Explanations

L1 human

A series converges when its running totals approach a fixed number. Adding more and more terms then changes the total less and less.

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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 sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinition
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  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

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Sources

  1. Web https://en.wikipedia.org/wiki/Convergent_series, Definition

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