Divergent Series/Examples/sin i n over n^2
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Example of Divergent Series
The complex series defined as:
- $\ds S = \sum_{n \mathop = 1}^\infty \dfrac {\sin i n} {n^2}$
is divergent.
Proof
\(\ds \cmod {\dfrac {\sin i n} {n^2} }\) | \(=\) | \(\ds \cmod {\dfrac {\map \exp {i \paren {i n} } - \map \exp {-i \paren {i n} } } {2 i n^2} }\) | Euler's Sine Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {\dfrac {\map \exp {-n} - \exp n} {2 n^2} }\) | ||||||||||||
\(\ds \) | \(>\) | \(\ds \dfrac {e^n - 1} {2 n^2}\) | ||||||||||||
\(\ds \) | \(\to\) | \(\ds \infty\) |
Hence the result.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $2 \ \text {(v)}$