Convergence of Generalized Sum of Complex Numbers/Corollary
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Corollary to Convergence of Generalized Sum of Complex Numbers
Let $\family {z_j}_{j \mathop \in I}$ be an $I$-indexed family of complex numbers.
That is, let $z_j \in \C$ for all $j \in I$.
Suppose that $\ds \sum_{j \mathop \in I} z_j$ converges to $z \in \C$.
Then $\ds \sum_{j \mathop \in I} \overline {z_j}$ converges to $\overline z$, where $\overline z$ denotes the complex conjugate of $z$.
Here, the $\sum$ denote generalized sums.
Proof
Using Convergence of Generalized Sum of Complex Numbers, one has:
- $\ds \sum_{j \mathop \in I} \map \Re {z_j} = \map \Re z$
- $\ds \sum_{j \mathop \in I} \map \Im {z_j} = \map \Im z$
Now, observe that, from the definition of complex conjugate:
\(\ds \overline z\) | \(=\) | \(\ds \map \Re z - i \map \Im z\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop \in I} \map \Re {z_j} - i \sum_{j \mathop \in I} \map \Im {z_j}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop \in I} \paren {\map \Re {z_j} - i \map \Im {z_j} }\) | Generalized Sum is Linear | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop \in I} \overline {z_j}\) |
$\blacksquare$