Suppose that \(f\) is holomorphic in an open set containing the closed unit disc, except for a pole at \(z_0\) on the unit circle. Let denote the the power series in the open disc. Show that (1) \(c_n \ne 0\) for all large enough \(n\)’s, and (2) \(\lim_{n\to\infty} \frac{c_n}{c_{n+1}} = z_0\).
The source omits the defining expression between “Let” and “denote”, so the coefficients \(c_n\) are never introduced. From the conclusions, the missing text defines \(c_n\) as the coefficients of the power series expansion of \(f\) in the open unit disc, \(f(z) = \sum_{n=0}^\infty c_n z^n\) for \(\abs{z} < 1\); the source does not supply this wording.