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Definition: Real Infinite Series, Partial Sums

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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Bibliography (further reading)

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