\(
\newcommand{\cat}[1] {\mathrm{#1}}
\newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})}
\newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}}
\newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}}
\newcommand{\betaReduction}[0] {\rightarrow_{\beta}}
\newcommand{\betaEq}[0] {=_{\beta}}
\newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}}
\newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}}
\newcommand{\groupMul}[1] { \cdot_{\small{#1}}}
\newcommand{\inv}[1] {#1^{-1} }
\newcommand{\bm}[1] { \boldsymbol{#1} }
\require{physics}
\require{ams}
\)
Math and science::Analysis::Tao::07. Series
The Root Test
Let \( \sum_{n=m}^{\infty}a_n \) be a series of real numbers and let [ \( \alpha = ? \) ].
- If \( \alpha < 1 \), then the series \( \sum_{n=m}^{\infty}a_n \) is absolutely convergent (and hence conditionally convergent).
- If \( \alpha > 1 \), then the series \( \sum_{n=m}^{\infty}a_n \) is not conditionally convergent (and hence is not absolutely convergent either).
- If \( \alpha = 1 \), this test does not assert any conclusion.
The famous Root Test.
