(related to Proposition: Monotony Criterion)


Let \(\sum_{k=0}^\infty x_k\) be an infinite series with non-negative terms \(x_n\ge 0\) for all \(n\in\mathbb N\) and let sequence \((s_n)_{n\in\mathbb N}\) of its partial sums \(s_n:=\sum_{k=0}^n x_k\) be bounded. Then the sequence \((s_n)_{n\in\mathbb N}\) is monotonically increasing, and it follows from the monotone convergence theorem that \((s_n)_{n\in\mathbb N}\) is convergent.


Let the infinite series be convergent. Then the sequence \((s_n)_{n\in\mathbb N}\) of its partial sums is convergent. It follows from the corresponding proposition that convergent sequences are bounded.

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