proof of Cauchy’s root test

If for all $n\\geq N$

\\sqrt[n]{a_{n}}<k<1

then

a_{n}<k^{n}<1.

Since $\\sum_{i=N}^{\\infty}k^{i}$ converges so does $\\sum_{i=N}^{\\infty}a_{n}$ by the comparison test. If $\\sqrt[n]{a_{n}}>1$ then by comparison with $\\sum_{i=N}^{\\infty}1$ the series is divergent. Absolute convergence in case of nonpositive $a_{n}$ can be proven in exactly the same way using $\\sqrt[n]{|a_{n}|}$ .

随便看

AlternativeCharacterizationOfUltrafilter
alternative characterization of ultrafilter
AlternativeCharacterizationsOfNoetherianTopologicalSpaces
alternative characterizations of Noetherian topological spaces
AlternativeCharacterizationsOfNoetherianTopologicalSpacesProofOf
alternative characterizations of Noetherian topological spaces, proof of
AlternativeCharacterizationsOfRecursiveFunctions
alternative characterizations of recursive functions
AlternativeDefinitionOfAlgebraicallyClosed
alternative definition of algebraically closed
AlternativeDefinitionOfAMultigraph
alternative definition of a multigraph
AlternativeDefinitionOfAQuasigroup
alternative definition of a quasigroup
AlternativeDefinitionOfCardinality
alternative definition of cardinality
AlternativeDefinitionOfGroup
alternative definition of group
AlternativeDefinitionOfKrullValuation
alternative definition of Krull valuation
AlternativeDefinitionOfLebesgueIntegralAn
alternative definition of Lebesgue integral, an
AlternativeDefinitionOfMetricSpace
alternative definition of metric space
AlternativeDefinitionOfTheNaturalLogarithm

数学辞典收录了18232条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。