Sine Function is Absolutely Convergent/Complex Case/Proof 1

Proof
The definition of the complex sine function is:


 * $\ds \forall z \in \C: \sin z = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}$

By definition of absolutely convergent complex series, we must show that the power series


 * $\ds \sum_{n \mathop = 0}^\infty \size {\paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!} }$

is convergent.

We have

The result follows from Squeeze Theorem for Sequences of Complex Numbers.