On the Function of Taylor Formula

In mathematics, Taylor formula is a formula that uses the information of a function at a certain point to describe the value near it. If the function is smooth enough, Taylor formula can use these derivative values as coefficients to construct a polynomial to approximate the value of the function at this point. Taylor formula also gives the deviation of this polynomial from the actual function value.

take for example

f(x)=f(x0)+f(x0)'(x-x0)+0(x-x0)

Approximate function f(x) of f(x0)+f('x0)(x-x0) at point x0.

But approximation is not enough.

Is to approximate a function with higher order.

Of course, we must also satisfy that the error is high-order infinitesimal.

So compare the above formula.

There are:

pn(x)=a0+a 1(x-x0)+a2(x-x0)^2+...+an(x-x0)^n

Where an = pn (n) (x0)/n!

Same form as above.

Finally, it is proved that the higher order is infinitely small!

I don't know. How's this?