Radius of Convergence of Power Series Expansion for Sine Function
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Theorem
The sine function has the complex power series expansion:
\(\ds S \paren z\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds z - \frac {z^3} {3!} + \frac {z^5} {5!} - \frac {z^7} {7!} + \cdots\) |
which is the power series expansion of the sine function.
This is valid for all $z \in \C$.
Proof
Applying Radius of Convergence from Limit of Sequence: Complex Case, we find that:
\(\ds \lim_{n \mathop \to \infty} \cmod {\dfrac {a_{n + 1} } {a_n} }\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \cmod {\dfrac {\frac {\paren {-1}^{n + 1} } {\paren {2 \paren {n + 1} + 1}!} } {\frac {\paren {-1}^n} {\paren {2 n + 1}!} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \cmod {\dfrac {\paren {2 n + 1}!} {\paren {2 \paren {n + 1} + 1}!} }\) | as $\cmod {\paren {-1}^n} = 1$ for all $n$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \cmod {\dfrac 1 {\paren {2 n + 2} \paren {2 n + 3} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | Sequence of Powers of Reciprocals is Null Sequence |
Hence the result.
$\blacksquare$
Also see
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.4$. Power Series: Example $\text {(iii)}$