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Cauchy's root test (Theorem)

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.

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.




"Cauchy's root test" is owned by Mathprof. [ full author list (2) | owner history (1) ]
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See Also: Lambert series

Other names:  root test

Attachments:
proof of Cauchy's root test (Proof) by mathwizard
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Cross-references: absolutely convergent, complex, divergent, number, infinite, convergent, terms, real, positive, series
There are 3 references to this entry.

This is version 6 of Cauchy's root test, born on 2002-08-23, modified 2006-10-24.
Object id is 3337, canonical name is CauchysRootTest.
Accessed 7734 times total.

Classification:
AMS MSC40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)

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