What is an interval, |S − S4| ≤ |R4| for which the actual sum exists?

Question

The series:  has a partial sum  when x = 1. What is an interval, |S − S₄| ≤ |R₄| for which the actual sum exists? Provide an exact answer and justify your conclusion. 

Answer 1

I answered a very similar question recently.

First of all this is the Taylor series for cos x which converges for all x.

It is an alternating sign series for which the error does not exceed in absolute value the first omitted term.

Consequently when x=1, the error is less than 1/9!.