Taylor Series

Suppose we have an unknown function $f(x)$, but we somehow have access to its derivatives at a certain point. We might then try to approximate the function near that point using a polynomial expansion.
The idea is that if our polynomial matches not only the value of $f$ but also some of its derivatives at that point, it can locally mimic the behavior of $f$ quite well.

Polynomial Approximations Around $x = 0$

Let’s start by approximating $f(x)$ around the point $x = 0$.

• Zeroth-order approximation (degree 0):

  $$ p(x) = f(0) $$

  This is a constant approximation — clearly very crude, but it matches $f(x)$ at $x = 0$.

• First-order approximation (degree 1):

  We now require that our polynomial $p(x)$ also matches the first derivative of $f$ at 0:

  $$ p(0) = f(0), \quad p’(0) = f’(0) $$

  The simplest polynomial satisfying these conditions is:

  $$ p(x) = f(0) + f’(0)x $$

  This gives us the tangent line to $f$ at $x = 0$.

• Second-order approximation (degree 2):

  We now add a quadratic term and require that $p’‘(0) = f’’(0)$:

  $$ p(x) = f(0) + f’(0)x + \frac{1}{2}f’’(0)x^2 $$

  Checking the derivatives:

  $$ p’(x) = f’(0) + f’‘(0)x \Rightarrow p’(0) = f’(0) $$

  $$ p’‘(x) = f’‘(0) \Rightarrow p’‘(0) = f’’(0) $$

  So this quadratic polynomial matches $f$, $f’$, and $f’’$ at $x = 0$.

The General Pattern

Continuing this process indefinitely gives the Maclaurin series, which is just a Taylor series expanded around $x = 0$:

$$ p(x) = f(0) + f’(0)x + \frac{f’’(0)}{2!}x^2 + \frac{f^{(3)}(0)}{3!}x^3 + \dots + \frac{f^{(n)}(0)}{n!}x^n + \dots $$

This provides an increasingly accurate local approximation of $f(x)$ as more terms are added.

Taylor Series Around an Arbitrary Point $x = c$

If instead we want to approximate $f(x)$ around some point $x = c$, we shift the expansion:

$$ p(x) = f(c) + f’(c)(x - c) + \frac{f’’(c)}{2!}(x - c)^2 + \frac{f^{(3)}(c)}{3!}(x - c)^3 + \dots $$

or more compactly:

$$ p(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!} (x - c)^n $$

This is the Taylor series of $f$ around $x = c$.
It represents how we can reconstruct the local behavior of $f$ using its derivatives, which will be crucial when we approximate derivatives numerically (e.g. in finite-difference schemes).