Convergence of Generalized Sum of Complex Numbers/Corollary

From ProofWiki
Jump to navigation Jump to search

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
\(\ds \) \(=\) \(\ds \sum_{j \mathop \in I} \overline {z_j}\)

$\blacksquare$