Math and science::Analysis::Tao::07. Series

Finite series, definition

Let

m, n

be integers, and let

(a_{i})_{i = n}^{m}

be a finite sequence of real numbers, assigning a real number

a_{i}

for each integer

i

between

n

and

m

inclusive (i.e.

m \leq i \leq n

). Then we define the finite sum (or finite series)

\sum_{i = m}^{n} a_{i}

by the recursive formula

\begin{aligned} \sum_{i = m}^{n} a_{i} & = 0 whenever n < m; \\ \sum_{i = m}^{n + 1} a_{i} & = (\sum_{i = m}^{n} a_{i}) + a_{n + 1} whenever n + 1 \geq m . \end{aligned}

The series is often less formally expressed as:

\sum_{n}^{i = m} a_{i} = a_{m} + a_{m + 1} + . . . + a_{n}

Semantically, "series" refers to the expression of the form

\sum_{n}^{i = m} a_{i}

, which is mathematically, but not semantically equal to a real number, which is called the "sum" of the series. This is a linguistic distinction, and a distinction that is not pertinant in mathematics due to the axiom of substitution.

Example

$\begin{aligned} \sum_{i = m}^{m - 2} a_{i} & = 0 \\ \sum_{i = m}^{m - 1} a_{i} & = 0 \\ \sum_{i = m}^{m} a_{i} & = a_{m} \\ \sum_{i = m}^{m + 1} a_{i} & = a_{m} + a_{m + 1} \\ \sum_{i = m}^{m + 2} a_{i} & = a_{m} + a_{m + 1} + a_{m + 2} \end{aligned}$

Source

p156-157

15.11.2019