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 $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ converges to $Z \in \C$

if and only if:

the series:
$\displaystyle \sum_{n \mathop = 1}^\infty \Re \paren {z_n}$
and:
$\displaystyle \sum_{n \mathop = 1}^\infty \Im \paren {z_n}$
converge to $\Re \paren Z$ and $\Im \paren Z$ respectively.


Proof

Let:

the $n$th partial sum of $\sequence {z_n}$ be denoted $Z_n$
the $n$th partial sum of $\sequence {\Re \paren {z_n} }$ be denoted $U_n$
the $n$th partial sum of $\sequence {\Im \paren {z_n} }$ 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.

$\blacksquare$


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