Given the following Taylor series:
We know that:
- It converges for all of $x$
- It converges to the function $\cos x$
The Taylor series converge for all of $x$ if for a fixed value of $x$, the partial sums converge to a limit, $L$.
The Taylor series converge to $\cos x$ if its error term is $0$ as $n$ (the number of terms in the Taylor series) goes to infinity.
My question is:
Are these two concepts related?
I thought point 1 would be useful when proving point 2. But when doing the proof of 2, I don't see any connection to point 1. If the two concepts are not related, then why is it useful to know the interval of convergence of a Taylor series (or any series for that matter)?