\(
\newcommand{\cat}[1] {\mathrm{#1}}
\newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})}
\newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}}
\newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}}
\newcommand{\betaReduction}[0] {\rightarrow_{\beta}}
\newcommand{\betaEq}[0] {=_{\beta}}
\newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}}
\newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}}
\newcommand{\groupMul}[1] { \cdot_{\small{#1}}}
\newcommand{\groupAdd}[1] { +_{\small{#1}}}
\newcommand{\inv}[1] {#1^{-1} }
\newcommand{\bm}[1] { \boldsymbol{#1} }
\require{physics}
\require{ams}
\require{mathtools}
\)
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
\[\begin{aligned}\sum_{i=m}^{n} a_i &= 0 \text{ whenever } n < m; \\\sum_{i=m}^{n+1} a_i &= [...] \text{ whenever } [...].\end{aligned}\]
