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Cauchy's root test

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If $\sum a_{n}$ is a series of positive real terms and
$$\sqrt[n]{a_{n}} < k < 1$$
for all $n > N$, then $\sum a_{n}$ is convergent.  If $\sqrt[n]{a_{n}} \geq 1$ for an infinite number of values of $n$, then $\sum a_{n}$ is divergent.

\subsubsection*{Limit form} Given a series $\sum a_{n}$ of complex terms, set
$$\rho = \limsup_{n \to \infty} \sqrt[n]{| a_{n} |}$$
The series $\sum a_{n}$ is absolutely convergent if $\rho < 1$ and is divergent if $\rho > 1$.  If $\rho = 1$, then the test is inconclusive.
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