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Definition: Convergent Real Series

A real series \(\sum_{k=0}^\infty x_k\) is called convergent, if 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 convergent. For convergent real series, the notation

\[\sum_{k=0}^\infty x_k\]

can, depending on the context, denote two things:

1. the convergent real series itself or
2. the convergent real series itself or

Table of Contents

1. Proposition: Cauchy Product of Convergent Series Is Not Necessarily Convergent
2. Proposition: Sum of Convergent Real Series
3. Proposition: Difference of Convergent Real Series
4. Proposition: Product of a Real Number and a Convergent Real Series

Mentioned in:

Definitions: 1 2 3
Lemmas: 4
Proofs: 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Propositions: 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

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References

Bibliography

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