What is Taylor's Inequality about?

Question

I am confused by the Taylor's Inequality formula: $$|R_n(x)| \le \frac M{(n+1)!}|x−a|^{n+1}$$

1. What do $R_n$, $M$, $n$, $x$, $a$, stand for i.e. what to substitute inside each variable?
2. What purpose does this formula serve/How to apply it in situations/problems?

Answer 1

$R_n(x)=f(x)-T_n(x)$ is the difference between the Taylor polynomial (the approximation) and the actual value of the function (the thing you want to approximate). You want it to be small.

$M$ is a bound on the magnitude of the $(n+1)$th derivative. You would like it to be small, but you don't really have control over it; it's a property of the given function and $x$.

$n$ is the degree of the Taylor polynomial. This is the thing you can change.

$x$ is the point where you are trying to evaluate $f(x)$.

$a$ is the point where the Taylor polynomial is expanded, i.e. the point where you are taking the derivatives. (Note that this is not the point where the derivative is evaluated in the remainder formula.)

Provided you can obtain a value of $M$ (the hard part) and it doesn't grow too fast with $n$, you can use this to determine how many terms of the Taylor polynomial are needed to obtain a given accuracy. For example, one can show that for the exponential function,

$$\frac{\exp(x)-T_n(x)}{\exp(x)} \leq \frac{x^{n+1}}{(n+1)!}$$

for $x>0$. This means that you can make the relative error small by making the right side small. In a calculus class you would probably do this by brute force checking a few values.

Answer 2

When a series converge, the partial sums are approximations of the limit.

How good an approximation they are depends on how small the difference between the limit and the partial sum is.

If you compute several terms and see that some digits in the partial sum don't seem to change after a while, it is not a proof that they won't change again. To prove it, what you need is exactly this kind of formula.