The study (iii)
The limit for the Padé approximants and exist for all . This is much stronger than the Taylor series, which only converges inside the unit disk and diverges outside.
On the line the limit does not exist for , , where the original function has a logarithmic branch cut. This fact is “captured” by the Padé rational functions by a funny effect: and have poles for which become dense in . E.g. :
Although divergent, in some instances the Padé’s still exhibit a rich structure. E.g. has two limit points at :
The esoteric conclusion: the Padé’s are much more efficient than the Taylor approximations and “somehow know” about the analytic properties of the original function.