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]

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[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