Divergent Series/Examples/((2+3i) over (3-2i))^n
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Example of Divergent Series
The complex series defined as:
- $\ds S = \sum_{n \mathop = 1}^\infty \paren {\dfrac {2 + 3 i} {3 - 2 i} }^n$
is divergent.
Proof
\(\ds \cmod {\paren {\dfrac {2 + 3 i} {3 - 2 i} }^n}\) | \(=\) | \(\ds \cmod {\dfrac {2 + 3 i} {3 - 2 i} }^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac {4 + 9} {9 + 4} }^{n/2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
Thus the sequence $\sequence {\paren {\dfrac {2 + 3 i} {3 - 2 i} }^n}$ does not tend to zero.
Hence from Terms in Convergent Series Converge to Zero it follows that $S$ is divergent.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.3$. Series: Example $\text{(ii)}$