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. \( m \le i \le n \)). Then we define the finite sum (or finite series) \( \sum_{i=m}^{n} a_i \) by the recursive formula\[\begin{aligned}\sum_{i=m}^{n} a_i &= 0 \text{ whenever } n < m; \\\sum_{i=m}^{n+1} a_i &= \left(\sum_{i=m}^{n} a_i\right) + a_{n+1} \text{ whenever } n + 1 \ge m.\end{aligned}\]
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
\[
\begin{aligned}
\sum_{i=m}^{m-2} a_i &= 0\\
\sum_{i=m}^{m-1} a_i &= 0\\
\sum_{i=m}^{m} a_i &= a_m\\
\sum_{i=m}^{m+1} a_i &= a_m + a_{m+1} \\
\sum_{i=m}^{m+2} a_i &= a_m + a_{m+1} + a_{m+2}\end{aligned}\]