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:

$S = \displaystyle \sum_{n \mathop \ge 0} a_n z^n$

has a radius of convergence of $\dfrac 1 e$.


Proof

Let $R$ denote the radius of convergence of $S$.

Thus:

\(\displaystyle R\) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \cmod {\dfrac {a_{n - 1} } {a_n} }\) Radius of Convergence from Limit of Sequence
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \cmod {\dfrac {\cos i \paren {n - 1} } {\paren {n - 1}^2} / \dfrac {\cos i n} {n^2} }\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \cmod {\dfrac {\cos i \paren {n - 1} } {\cos i n} \dfrac {n^2} {\paren {n - 1}^2} }\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \cmod {\dfrac {\exp \paren {i \paren {i \paren {n - 1} } } + \exp \paren {-i \paren {i \paren {n - 1} } } } {\exp \paren {i \paren {i n} } + \exp \paren {-i \paren {i n} } } \dfrac 1 {\paren {1 - \frac 1 n}^2} }\) Cosine Exponential Formulation and simplifying
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \cmod {\dfrac {\exp \paren {-\paren {n - 1} } + \exp \paren {n - 1} } {\exp \paren {-n} + \exp \paren n} } \cmod {\dfrac 1 {\paren {1 - \frac 1 n}^2} }\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\exp \paren {n - 1} } {\exp \paren n}\) as $\sequence {\dfrac 1 n}$ is a basic null sequence and $e^{-n} \to 0$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\exp \paren n \exp \paren {-1} } {\exp \paren n}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 e\)

$\blacksquare$


Sources