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Definition: Convergent Complex Sequence

A convergent complex sequence is a complex sequence \((x_n)_{n\in\mathbb N}\), which is convergent in the metric space of complex numbers \((\mathbb C,|~|)\). In other words, \((x_n)_{n\in\mathbb N}\) is convergent to the number \(x\in\mathbb C\), if and only if for each \(\epsilon > 0\) there exists an \(N\in\mathbb N\) with \[ | x_n-x | < \epsilon\quad\quad \text{ for all }n\ge N.\]

If \((x_n)_{n\in\mathbb N}\) is convergent to the number \(x\in\mathbb C\), we write \[\lim_{n\to\infty} x_n=x.\]

Table of Contents

1. Proposition: Quotient of Convergent Complex Sequences
2. Proposition: Sum of Convergent Complex Sequences
3. Proposition: Difference of Convergent Complex Sequences
4. Proposition: Product of Convegent Complex Sequences
5. Proposition: Product of a Complex Number and a Convergent Complex Sequence

Mentioned in:

Corollaries: 1
Definitions: 2 3 4
Proofs: 5 6 7 8 9 10 11 12 13 14 15
Propositions: 16 17 18 19 20 21 22 23 24 25 26
Theorems: 27

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References

Bibliography

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