Let \((x_n)_{n\in\mathbb N}\) be a real sequence. The real sequence \((s_n)_{n\in\mathbb N}\) of **partial sums**

\[s_n:=\sum_{k=0}^n x_k,\quad\quad n\in\mathbb N\]

is called the **(infinite) real series**

\[\sum_{k=0}^\infty x_k\quad\quad( * ).\]

Note: If the sequence of partial sums is convergent, the expression \( ( * ) \) also denotes the limit to which the sequence converges.

| | | | | created: 2015-02-18 14:20:32 | modified: 2019-09-01 13:32:20 | by: *bookofproofs* | references: [581]

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