Complex Power Series/Examples/cos i n over n^2
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Example of Complex Power Series
Let $\sequence {a_n}$ be the sequence defined as:
- $a_n = \dfrac {\cos i n} {n^2} z^n$
The complex power series:
- $S = \ds \sum_{n \mathop \ge 0} a_n z^n$
has a radius of convergence of $\dfrac 1 e$.
Proof
Let $R$ denote the radius of convergence of $S$.
Thus:
\(\ds R\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \cmod {\dfrac {a_{n - 1} } {a_n} }\) | Radius of Convergence from Limit of Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \cmod {\dfrac {\cos i \paren {n - 1} } {\paren {n - 1}^2} / \dfrac {\cos i n} {n^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \cmod {\dfrac {\cos i \paren {n - 1} } {\cos i n} \dfrac {n^2} {\paren {n - 1}^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \cmod {\dfrac {\exp \paren {i \paren {i \paren {n - 1} } } + \exp \paren {-i \paren {i \paren {n - 1} } } } {\exp \paren {i \paren {i n} } + \exp \paren {-i \paren {i n} } } \dfrac 1 {\paren {1 - \frac 1 n}^2} }\) | Euler's Cosine Identity and simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \cmod {\dfrac {\exp \paren {-\paren {n - 1} } + \exp \paren {n - 1} } {\exp \paren {-n} + \exp \paren n} } \cmod {\dfrac 1 {\paren {1 - \frac 1 n}^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\exp \paren {n - 1} } {\exp \paren n}\) | as $\sequence {\dfrac 1 n}$ is a basic null sequence and $e^{-n} \to 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\exp \paren n \exp \paren {-1} } {\exp \paren n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 e\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $3 \ \text {(iv)}$