Proposition: Leibniz Criterion for Alternating Series

Let \((a_n)_{n\in\mathbb N}\) be a monotonically decreasing real sequence, which is convergent to \(0\), i.e. with \(\lim_{n\to\infty} a_n=0\). Then the alternating series \[\sum_{n=0}^\infty (-1)^n a_n\] is a convergent series. This convergence criterion is also known as the Leibniz criterion and was first proven by the Gottfried Wilhelm von Leibniz (1646 - 1716).

Proofs: 1

Proofs: 1 2

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983