In the book 'The Concrete Tetrahedron' by Manuel Kauers and Peter Paule on p30 reads or implies the function $$f: C\backslash \{ ... ,-2\pi,-\pi,0,\pi,2\pi, ...\}\rightarrow C,\, f(z)=\frac z{\sin(z)}$$(where I have used $C$ to mean complex numbers) has a removable singularity at $z\!=\!0$, and the extension $f(0):=1$ turns it into a function that is analytic in a neighborhood of the origin. What exactly is meant by the extension $f(0):=1$ ? Could anyone explain it in different words ? If left off the $f: C\backslash \{ ... ,-2\pi,-\pi,0,\pi,2\pi, ...\}\rightarrow C$ part , that is just plain $\frac z{\sin(z)}$, does not have a singularity at $z\!=\!0$ and has radius of convergence $r\!=\!\pi$ ? Is adding the extension $f(0):=1$ to $$f: C\backslash \{ ... ,-2\pi,-\pi,0,\pi,2\pi, ...\}\rightarrow C,\, f(z)=\frac z{\sin(z)}$$ essentially the same as $\frac z{\sin(z)}$ within $r\!=\!\pi$ ?
On p31 it reads or implies coefficients of the formal power series $$\frac x {\sin(x)}-(\frac \pi {\pi-x}-\frac\pi{-\pi-x} ) $$ are bounded in absolute value by $(\frac1{2\pi}+\eta)^n$ when n is large. In other words $$|a_n-\pi^{-n}+({-\pi})^{-n}|<(\frac1{2\pi}+\eta)^n,\,n\!\rightarrow\!\infty$$ for every $\eta>0$, it follows that $a_{2n}\sim 2\pi^{-2n}$ as $n\!\rightarrow\!\infty.$ Is it a text error and it should have been written $a_{2n}\sim (2\pi)^{-2n}$ instead of $a_{2n}\sim 2\pi^{-2n}$ ? Is that correct ? Next suppose instead it were coefficients of the formal power series $$\frac x {\sin(x)}-(\frac \pi {\pi-x}-\frac\pi{-\pi-x} )-(\frac{2\pi} {2\pi-x}-\frac{2\pi}{-2\pi-x}) $$ then would $a_{2n}\sim (3\pi)^{-2n}$ ?
For the extension problem, consider the following simpler case.
Let $g:\mathbb{C}\setminus\{0\}$ be $g(z)=\frac{\sin z}{z}$. Define a function $h:\mathbb{C}\to\mathbb{C}$ with $h(z)=g(z)$ for $z\ne 0$ and $h(0)=1$. Then the function $h$ is called an extension of $g$. Moreoever, $h$ is analytic on $\mathbb{C}$, particularly around $z=0$.
The asymptomatic expression should indeed be $a_{2n}\sim(2\pi)^{-2n}$ since the constant matters by the definition on page 165 of the book.
For your last question, the answer is yes. Because your new function, denoted it as $g$, has removable singularities at $0$, $\pm \pi$, $\pm 2\pi$; $g$ can thus be extended to an analytic function in a disk of radius $3\pi$ around the origin. Now you can follow the same logic for the function on page 31.