MATH 144: Calculus for the Physical Sciences I

Concept 7.4.1. Taylor polynomials.

The idea of Taylor polynomials is to approximate a function $f(x)$ at a point $x=a$ by higher degree polynomials. The polynomial

where $k!= 1 \cdot 2 \cdot 3 \dots k\text{,}$ is called the degree $d$ Taylor polynomial of $f(x)$ at $x = a\text{.}$

It satisfies the properties:

for all $k=1,2,\dots, d\text{.}$

Concept 7.4.2. The degree $d$ polynomial approximation of a function at a point.

The degree $d$ approximation of $f(x)$ at $x = a$ is

where $T_d(x)$ is the degree $d$ Taylor polynomial of $f(x)$ at $x=a\text{.}$ When $d=1$ we get the linear approximation of $f$ at $x=a\text{,}$

When $d=2$ we get the quadratic approximation of $f$ at $x=a\text{,}$

Concept 7.4.3. The Taylor series.

In fact, we could try to send $d \to \infty\text{;}$ that would give an “approximation” of $f(x)$ at $x=a$ by an “infinite degree polynomial”. This can be made rigorous using sequence and series; namely, taking $d\to \infty$ means taking the limit of the sequence of Taylor polynomials $T_d(x)\text{,}$ and the result is the Taylor series of $f(x)$ at $x=a$. Reversing the process, one can think of the Taylor polynomials above as truncations (so-called “partial sums”) of the Taylor series of $f(x)$ at $x=a\text{.}$