Math 2.0

definition

Sum of a series

Definition

Let ∑an\sum a_n be a convergent series of real numbers with partial sums sN=∑n=0Nans_N = \sum_{n=0}^{N} a_n. The sum of the series is the real number ∑n=0∞an=lim⁡N→∞sN.\sum_{n=0}^{\infty} a_n = \lim_{N \to \infty} s_N.

Remarks

The sum is well defined because a sequence has at most one limit. The sum is linear: ∑(an+bn)=∑an+∑bn\sum (a_n + b_n) = \sum a_n + \sum b_n and ∑c an=c∑an\sum c\, a_n = c \sum a_n when ∑an\sum a_n and ∑bn\sum b_n converge. This follows from the algebra of limits. For example, ∑n=0∞1/2n=2\sum_{n=0}^{\infty} 1/2^n = 2.

Explanations

L1 human

The sum of a series is the number that its running totals approach. For 1+1/2+1/4+⋯1 + 1/2 + 1/4 + \cdots, the totals 1,1.5,1.75,1.875,…1, 1.5, 1.75, 1.875, \dots approach 22, so the sum is 22.

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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 seriesdefinitionSum of a series, definition, no evidence levelSum of a seriesdefinition
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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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
  8. Sum of a series definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Series_(mathematics), Sum of a series

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