Root Test (Cauchy)

Definition

$\rho$ (rho): Greek letter, pronounced “row”, denotes the limit of $n$ -th roots.

Root test

For a positive series $\sum_{n=1}^{\infty} a_n$ with $a_n \ge 0$ , if

$\lim_{n \to \infty} \sqrt[n]{a_n} = \rho,$

then:

Root test rule

$\lim_{n \to \infty} \sqrt[n]{a_n} = \rho$

符号	类型	读音/说明	在本文中的含义
$\rho$	希腊字母	Rho（柔）	表示级数收敛性判别中的极限值
$\sum$	希腊字母	Sigma（西格玛）	求和符号，表示级数
$\infty$	数学符号	无穷大	表示无穷级数，项数无限
$\lim$	数学符号	极限	表示数列或函数的极限
$\sqrt[n]{a_n}$	数学符号	$n$ 次根	$a_n$ 的 $n$ 次根

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