\(
\newcommand{\cat}[1] {\mathrm{#1}}
\newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})}
\newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}}
\newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}}
\newcommand{\betaReduction}[0] {\rightarrow_{\beta}}
\newcommand{\betaEq}[0] {=_{\beta}}
\newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}}
\newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}}
\newcommand{\groupMul}[1] { \cdot_{\small{#1}}}
\newcommand{\groupAdd}[1] { +_{\small{#1}}}
\newcommand{\inv}[1] {#1^{-1} }
\newcommand{\bm}[1] { \boldsymbol{#1} }
\require{physics}
\require{ams}
\require{mathtools}
\)
Math and science::Analysis::Tao::07. Series
Alternating series test
An alternating series converges if the sequence of elements converges absolutely to zero.
Let \( (a_n)_{n=m}^{\infty} \) be a sequence of real numbers which are non-negative and decreasing, thus \( a_n \ge 0 \) and \( a_n \ge a_{n+1} \) for every \( n \ge m \). Then the series \( \sum_{n=m}^{\infty}(-1)^n a_n \) is convergent if and only if the sequence [...] converges to 0 as \( n \rightarrow \infty \).
