# Convergent Complex Series/Examples/((2+3i) over (4+i))^n

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## Example of Convergent Complex Series

The series $\ds \sum_{n \mathop = 1}^\infty a_n$, where:

$a_n = \paren {\dfrac {2 + 3 i} {4 + i} }^n$

is convergent.

## Proof

 $\ds \cmod {\dfrac {2 + 3 i} {4 + i} }$ $=$ $\ds \cmod {\dfrac {\paren {2 + 3 i} \paren {4 - i} } {\paren {4 + i} \paren {4 - i} } }$ $\ds$ $=$ $\ds \cmod {\dfrac {8 + 10 i + 3} {4^2 + 1^2} }$ $\ds$ $=$ $\ds \cmod {\dfrac {11 + 10 i} {17} }$ $\ds$ $=$ $\ds \sqrt {\dfrac {11^2 + 10^2} {17^2} }$ Definition of Complex Modulus $\ds$ $=$ $\ds \sqrt {\dfrac {221} {289} }$ Definition of Complex Modulus $\ds$ $<$ $\ds 1$

Thus $\ds \sum_{n \mathop = 1}^\infty \paren {\dfrac {2 + 3 i} {4 + i} }^n$ is absolutely convergent by Sum of Infinite Geometric Sequence.

The result follows from Absolutely Convergent Series is Convergent: Complex Numbers.

$\blacksquare$