Taylor series refresher

This page is a beginner-friendly refresher for experiments that use Taylor polynomials. You only need basic calculus (derivatives) to follow it.

Taylor polynomial

Assume \(f\) has enough derivatives near \(x_0\) (this is true for \(\sin(x)\), \(\cos(x)\), polynomials, exponentials, etc.). The Taylor polynomial of degree \(n\) around \(x_0\) is

\[ T_n(x; x_0) = \sum_{k=0}^{n} \frac{f^{(k)}(x_0)}{k!}\,(x-x_0)^k . \]

Intuition: \(T_n(x;x_0)\) is the polynomial that matches \(f(x_0)\) and the first \(n\) derivatives at \(x_0\). It is usually accurate when \(x\) is close to \(x_0\).

For \(f(x)=\sin(x)\), the derivatives cycle, and around \(x_0=0\) this becomes

\[ \sin(x) \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots . \]

Truncation error and the remainder

The approximation error is the remainder

\[ R_n(x; x_0) = f(x) - T_n(x; x_0). \]
Under standard conditions, Taylor’s theorem gives a remainder representation. One common form is the (Lagrange) remainder:
\[ R_n(x; x_0) = \frac{f^{(n+1)}(\xi)}{(n+1)!}\,(x-x_0)^{n+1} \quad \text{for some } \xi \text{ between } x \text{ and } x_0 . \]
This is the key qualitative message for experiments like E001:

  • the factor \((x-x_0)^{n+1}\) makes the method local (good near \(x_0\), potentially bad far away),

  • increasing \(n\) helps most where \(|x-x_0|\) is small.

What experiments typically visualize

In a numerical experiment, you often look at

  • absolute error: \(|R_n(x; x_0)|\)

  • relative error: \(|R_n(x; x_0)| / |f(x)|\) (careful near zeros of \(f\))

and plot them across a domain to see where the approximation is reliable.

Practical numerical caveats

Even when the mathematics are correct, computation can mislead:

  • large \(|x-x_0|\) and high \(n\) can produce huge intermediate terms,

  • subtractive cancellation can reduce accuracy,

  • floating-point rounding can dominate before the theoretical truncation error does.

A common “extension” experiment is to repeat the same plots using higher precision arithmetic to separate truncation error from rounding error.

Introductory reading

If you want a longer, beginner-friendly treatment (beyond this refresher), these are good starting points:

References

See References.

[Apostol, 1991, Burden et al., 2015, Spivak, 2008, Wikipedia contributors, 2025]