Definition: Divergent Series

A real series $\sum_{k=0}^\infty x_k$ is called divergent, if it is not convergent. This is equivalent to the statement that the real sequence $(s_n)_{n\in\mathbb N}$ of partial sums $$s_n:=\sum_{k=0}^n x_k,\quad\quad n\in\mathbb N$$ tends to infinity, (respectively tends to minus infinity) . In this case, we write $$\sum_{k=0}^\infty x_n =\infty\quad\text{(respectively }\sum_{k=0}^\infty x_n =-\infty\text{)}.$$

Proofs: 1 2 3 4 5 6 7
Propositions: 8 9 10 11 12 13 14

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