Can we find a sequence, say p_j(z) such that p_j ---> 1/z uniformly for z is an element of an annulus between 1 and 2, that is 1 < abs(z) < 2?
Then i am asked to do the same thing but for p_j ---> sin(1/z^2).
Not too sure about this, maybe Taylor series/Laurent series expansions.
The Attempt at a Solution
So while I have no definite path yet set on proving this what I do have are a few thoughts. On this annulus 1/z is analytic because the point z = 0 is not contained. Also, we can write a Taylor/Laurent series expansion for 1/z.
However, I do not believe that we are able to do the same thing for the sin sequence because we end up with a larger numerator term which blows up and cause the series to diverge. However, against z = 0 is not contained here so maybe that is false? Am I thinking about this correctly or am I a fool?