Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 8 / c

For a nonterminating parity series, consecutive nonzero terms satisfyThe ratio test therefore gives $\boxed{R=1}$ for each nonterminating series. The nearest singular coefficient points are $x=\pm 1$, both distance one from the expansion centre.

The polynomial exception is essential here. If $n$ is an integer, the series of the same parity as $|n|$ terminates and has $\boxed{R=\infty }$; the opposite-parity independent series remains nonterminating with radius one. For noninteger $n$, both have radius one. A regular singular point can be removable for one particular solution, so its presence does not force every individual solution to have finite convergence radius.