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Proposition: Cauchy Condensation Criterion

An infinite series $\sum_{k=0}^\infty x_k$ with a monotonically decreasing real sequence $(x_k)_{k\in\mathbb N}$ of non-negative members $x_k\ge 0$ for all $k\in\mathbb N$ is convergent if and only if the “condensed series” $$\sum_{n=0}^\infty 2^n x_{2^n}$$ is convergent.

| | | | | created: 2020-01-26 11:01:49 | modified: 2020-02-01 06:19:24 | by: bookofproofs | references: [581], [586]

1.Proof: (related to "Cauchy Condensation Criterion")

2.Example: Applications of the Cauchy Condensation Criterion

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983

[586] Heuser Harro: “Lehrbuch der Analysis, Teil 1”, B.G. Teubner Stuttgart, 1994, 11. Auflage