Errors in Approximation, The Peano Remainder

As we said, the Taylor or Maclaurin Polynomials are ways of approximating the value of a function around a certain point using its derivatives. Of course, an approximation isn't perfect. There's going to be "too much" or "too little", so , some part that remains. A remainder.

This is the approximation error:

For a function with a Taylor Polynomial , the approximation error, how close or off this polynomial is in regards to the actual function it's approximating.

Which brings us to the Taylor Polynomial with Peano's Remainder:

(This is just the name of the "leftover" part during approximation)

Suppose you have a function f(x), differentiable n-times, at least once, in a neighborhood of a point a. If is the n-th order Taylor Polynomial of f(x) around that point a, then

We can also "turn" this theorem "inside-out":

Assume that you have some kind of polynomial of n-th order

Taylor and Maclaurin Polynomials in Function Analysis