Given the following Taylor series:
$1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}- \dots$
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)?
Best Answer
When you say that a series converges you are guaranteeing that you can keep it within some range you define. When you say that it converges to a function, then you are saying that within any error bounds you can choose enough terms of the sequence to get within the desired error bounds. When you are proving that the sequence converges to a specific function you can assume that it is that function and then find the desired number of terms needed to converge to that function. Basically, #1 is useful for proving #2 because you know it does converge so you can try to find what it converges to. The interval of convergence of a Taylor series is important because you can only use the Taylor series to approximate the function in the interval of convergence, which is why you use Taylor series. If you try to use the Taylor series outside of this interval your series won't converge.