I'm reading "An Introduction to Computational Physics" by Tao Pang. In it, he writes the following. In general, if the side length of a regular inscribed $k$-sided polygon is denoted as $l_k$ and the corresponding diameter is taken to be the unit of length, then the approximation of π is given by:
\begin{equation} \pi_k=kl_k \end{equation}
The exact value of $\pi$ is the limit of $\pi_k$ as $k\rightarrow \infty$. The value of $\pi_k$ obtained from the calculations of the k-sided polygon can be formally written as \begin{equation} \pi_{k}=\pi_{\infty}+\frac{c_{1}}{k}+\frac{c_{2}}{k^{2}}+\frac{c_{3}}{k^{3}}+\cdots \end{equation} where $\pi_\infty = π$ and $c_i$ , for $i = 1, 2, . . . ,\infty,$ are the coefficients to be determined.
The last part is the one I don't understand. Why is this form of expansion allowed and how is this the formal way to do it? Is it because of the undetermined coefficients? To my mind, you can expand anything in a series like that if you include undetermined coefficients.