Assumptions? Quasi-proof?

Can we assume that if for all values of , then for all ? Somehow I don't think this is valid, but let's run with it for a moment.

If this is true, then the whole point of the two-part exercise was to make the connection that for some where

Multiplying gives us . Now reduce the term and because the LHS is a whole number, the RHS must reduce fully. Since , 19 will still be a factor in this term after reducing, so it can be written for some positive integer k, which implies your hypothesis.

Not sure if this quasi-proof holds water, but it is the only connection I see between the two parts of the question.

Infinitely Differentiable

I think I worked it out in my head now. One of the stipulations of the proof is that the function expressible by the series is *infinitely differentiable* in the interval of convergence. So defining , could not possibly be equal to zero within some interval and nonzero at a point outside this interval. Otherwise the nth derivative of h at would have a left hand limit of zero and a nonzero right hand limit for some .

*Not trying to argue with mathematical proof, I'm just trying to get my intuition to catch up to it. Lemma: No infinitely differentiable function can be constant on an interval no matter how small, unless it is constant everywhere.