convergent series

A series $\\sum a_{n}$ is said to be convergentif the sequence of partial sums $\\sum_{i=1}^{n}a_{i}$ is convergent.A series that is not convergent is said to be divergent.

A series $\\sum a_{n}$ is said to be absolutely convergentif $\\sum|a_{n}|$ is convergent.

When the terms of the series live in $\\mathbb{R}^{n}$ ,an equivalent condition for absolute convergence of the seriesis that all possible series obtained by rearrangementsof the terms are also convergent.(This is not true in arbitrary metric spaces.)

It can be shown that absolute convergence implies convergence.A series that converges, but is not absolutely convergent,is called conditionally convergent.

Let $\\sum a_{n}$ be an absolutely convergent series,and $\\sum b_{n}$ be a conditionally convergent series.Then any rearrangement of $\\sum a_{n}$ is convergent to the same sum.It is a result due to Riemann that $\\sum b_{n}$ can be rearranged to converge to any sum, or not converge at all.

单词	ConvergentSeries
释义	convergent series A series $\\sum a_{n}$ is said to be convergentif the sequence of partial sums $\\sum_{i=1}^{n}a_{i}$ is convergent.A series that is not convergent is said to be divergent. A series $\\sum a_{n}$ is said to be absolutely convergentif $\\sum\|a_{n}\|$ is convergent. When the terms of the series live in $\\mathbb{R}^{n}$ ,an equivalent condition for absolute convergence of the seriesis that all possible series obtained by rearrangementsof the terms are also convergent.(This is not true in arbitrary metric spaces.) It can be shown that absolute convergence implies convergence.A series that converges, but is not absolutely convergent,is called conditionally convergent. Let $\\sum a_{n}$ be an absolutely convergent series,and $\\sum b_{n}$ be a conditionally convergent series.Then any rearrangement of $\\sum a_{n}$ is convergent to the same sum.It is a result due to Riemann that $\\sum b_{n}$ can be rearranged to converge to any sum, or not converge at all.
随便看	Simpson’s paradox Simpson’s rule SimsonsLine Simson’s line SimultaneousBlockdiagonalizationOfUpperTriangularCommutingMatrices simultaneous block-diagonalization of upper triangular commuting matrices SimultaneousConvergingOrDivergingOfProductAndSumTheorem simultaneous converging or diverging of product and sum theorem SimultaneousTriangularisationOfCommutingMatricesOverAnyField simultaneous triangularisation of commuting matrices over any field SimultaneousUpperTriangularBlockdiagonalizationOfCommutingMatrices simultaneous upper triangular block-diagonalization of commuting matrices SincIsL2 sinc is L^2 SincIsNotL1 sinc is not L^1 SineIntegral sine integral SineIntegralAtInfinity sine integral at infinity SineOfAngleOfTriangle sine of angle of triangle SinesLaw sines law SinesLawProof