Use of geometric series for Laurent series and domain of convergence for $\frac{1}{\sin (z)}$

Question

There has been a lot of explanation concerning the Laurent expansion of, for exemple,$\frac1{\sin(z)}$. The most common derivation I have encountered is: $$ \begin{align} \frac1{\sin(z)} &=\frac1z\frac{z}{\sin(z)}\\ &=\frac1z\left(1-\frac{z^2}{3!}+\frac{z^4}{5!}-\frac{z^6}{7!}+\cdots\right)^{-1}\\ &=\frac1z\left(1+\frac{z^2}{6}+\frac{7z^4}{360}+\frac{31z^6}{15120}+\cdots\right)\\ &=\frac1z+\frac{z}{6}+\frac{7z^3}{360}+\frac{31z^5}{15120}+\cdots \end{align} $$ However I find strange that it is not specified wether or not the term $-\frac{z^2}{3!}+\frac{z^4}{5!}-\frac{z^6}{7!}+\cdots$ is of modulus $\leq1$, in order to make proper use of the geometric series. I understand that in a neighbourhood of $0$, this term will satisfy the condition as it is holomorphic. However, I do not see how it can be used beyond a certain radius, yet I have often seen use of this derivation for similar problems, is there something I am missing?

Answer 1

You are right, for the geometric series we need that $\bigl\lvert \frac{\sin z}{z} - 1\bigr\rvert < 1$. And for the reordering and regrouping of $\sum_k (z^2/6 - z^4/120 + \dotsc)^k$, it is convenient to require

$$\sum_{n = 1}^{\infty} \frac{\lvert z\rvert^{2n}}{(2n+1)!} < 1$$

so that everything is clearly absolutely convergent and the manipulations legitimate.

As you say, in a small enough neighbourhood of $0$ that is the case, say for $\lvert z\rvert < r_0$.

Thus we have an expansion of $\frac{1}{\sin z}$ into a Laurent series that we know is valid in the annulus $0 < \lvert z\rvert < r_0$. Now we use the uniqueness of Laurent series. From the general theory, we know that $\frac{1}{\sin z}$ has a Laurent expansion

$$\frac{1}{\sin z} = \sum_{n = -\infty}^{+\infty} a_n z^n$$

that is valid in the annulus $0 < \lvert z\rvert < \pi$. In the annulus $0 < \lvert z\rvert < r_0$ (since $r_0 < \sqrt{6} < \pi$), we have two Laurent series representing the same function, hence by the uniqueness, these series are the same (have the same coefficients). Therefore, although the calculations were a priori only justified for $\lvert z\rvert < r_0$, it follows a posteriori that the series we obtained represents the function in the whole annulus $0 < \lvert z\rvert < \pi$.