Convergence of Series of Complex Numbers by Real and Imaginary Part
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Theorem
Let $\sequence {z_n}$ be a sequence of complex numbers.
Then:
- the series:
- $\ds \sum_{n \mathop = 1}^\infty \map \Re {z_n}$
- and:
- $\ds \sum_{n \mathop = 1}^\infty \map \Im {z_n}$
- converge to $\map \Re Z$ and $\map \Im Z$ respectively.
Proof
Let:
- the $n$th partial sum of $\sequence {z_n}$ be denoted $Z_n$
- the $n$th partial sum of $\sequence {\map \Re {z_n} }$ be denoted $U_n$
- the $n$th partial sum of $\sequence {\map \Im {z_n} }$ be denoted $V_n$
Then:
- $Z_n = U_n + i V_n$
Let:
- $\lim_{n \mathop \to \infty} U_n = U$
- $\lim_{n \mathop \to \infty} V_n = V$
By definition of convergent complex sequence:
- $\lim_{n \mathop \to \infty} Z_n = \lim_{n \mathop \to \infty} U_n + i \lim_{n \mathop \to \infty} V_n$
and so:
- $\lim_{n \mathop \to \infty} Z_n = U + i V$
and the result follows.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.3$. Series