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

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.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $2 \ \text {(iv)}$