Cropping off the Taylor Series

Question

We know that the Taylor series is for expansion of any function, but for digitization we need to crop off some parts? How can we determine upto which derivative should we consider..

I am mainly concerned regarding the Fourier series.

Answer 1

You can use the La'grange Remainder theorem to determine the possible range of error at each derivative you go up to.

From this tutorial:

Theorem 10.1 Lagrange Error Bound  

Let $f$ be a function such that it and all of its derivatives are continuous. If $P_{n}(x)$ is the nth Taylor polynomial for $f(x)$ centered at $x=a$, then the error is bounded by $|E_{n}(x)|\le\frac{M}{(n+1)!}|x-a|^{{n+1}}$. where M is some value satisfying $|f^{{(n+1)}}(x)|\le M$ on the interval between $a$ and $x$

If you want an error below .0001 for example, then you put in .0001 for $|E_{n}(x)|$ in the formula, and then solve for n.

Here is another example you can follow.