Math 2.0

definition

Series

Definition

Let (an)n∈N(a_n)_{n \in \mathbb{N}} be a real sequence. The series with terms ana_n is the sequence (sN)N∈N(s_N)_{N \in \mathbb{N}} given by sN=a0+a1+⋯+aN=∑n=0Nan.s_N = a_0 + a_1 + \cdots + a_N = \sum_{n=0}^{N} a_n. The notation for the series is ∑n=0∞an\sum_{n=0}^{\infty} a_n, or ∑an\sum a_n.

Remarks

The number sNs_N is a partial sum of the series. The symbol ∑n=0∞an\sum_{n=0}^{\infty} a_n also denotes the sum of the series, when the series is convergent. The context shows which meaning is intended. A series can start at another index, for example ∑n=1∞1/n\sum_{n=1}^{\infty} 1/n.

Explanations

L1 human

A series is an infinite sum a0+a1+a2+⋯a_0 + a_1 + a_2 + \cdots. We cannot add infinitely many numbers directly, so we add the first terms and watch what happens as we add more.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSeries, definition, no evidence levelSeriesdefinition
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  1. Sequence definition
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Sources

  1. Web https://en.wikipedia.org/wiki/Series_(mathematics), Definition

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