Conditional Convergence

conditional convergence

[kən′dish·ən·əl kən′vər·jəns] (mathematics) The property of a series that is convergent but not absolutely convergent.

Conditional Convergence

a concept of mathematical analysis. The series

is said to be conditionally convergent if it is convergent and the series

(whose terms are the absolute values of the terms of the original series) is divergent.

For example, the series

is conditionally convergent, since the absolute values of its terms form the divergent, harmonic series

If a series is conditionally convergent, then the two series consisting of its positive and negative terms, respectively, are divergent. According to Riemann’s theorem, by an appropriate rearrangement of the terms of a given conditionally convergent series we can obtain a divergent series or a series that has a prescribed sum. If two conditionally convergent series are multiplied term by term, a divergent series may result.

The concept of conditional convergence can be extended to series of vectors, to infinite products, and to improper integrals.

单词	conditional convergence
释义	conditional convergence conditional convergence n. Convergence of an infinite series that lacks absolute convergence, such as ¹/₂ + ¹/₃ + ¹/₄ .... Conditional Convergence conditional convergence [kən′dish·ən·əl kən′vər·jəns] (mathematics) The property of a series that is convergent but not absolutely convergent. Conditional Convergence a concept of mathematical analysis. The series is said to be conditionally convergent if it is convergent and the series (whose terms are the absolute values of the terms of the original series) is divergent. For example, the series is conditionally convergent, since the absolute values of its terms form the divergent, harmonic series If a series is conditionally convergent, then the two series consisting of its positive and negative terms, respectively, are divergent. According to Riemann’s theorem, by an appropriate rearrangement of the terms of a given conditionally convergent series we can obtain a divergent series or a series that has a prescribed sum. If two conditionally convergent series are multiplied term by term, a divergent series may result. The concept of conditional convergence can be extended to series of vectors, to infinite products, and to improper integrals.
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