Taylor Series

Computers can only add and multiply, so they approximate functions like $\sin x$ and $e^x$ with polynomials. Each extra term hugs the true curve over a wider range.

How it works

Power series approximation

In one sentence

A smooth function equals its Taylor series $\sum f^{(k)}(a)(x - a)^k/k!$ where the series converges; the degree- $n$ polynomial is the best polynomial approximation near $a$ .

General formula

{\htmlData{sym=3}{f(x)}} \approx \sum_{k=0}^{\htmlData{sym=4}{{n}}}\frac{{\htmlData{sym=0}{f^{(k)}(a)}}}{{\htmlData{sym=2}{k!}}}\,{\htmlData{sym=1}{(x - a)^k}}

Each term matches one more derivative at $a$ : the value, then the slope, then the curvature, and so on.

What the symbols mean

$f^{(k)}(a)$
k-th derivative at the center
a is the center the approximation is built around. This is how f bends there, measured k derivatives deep.
$(x - a)^k$
Distance from the center
How far x is from the center a, raised to the k-th power.
$k!$
Factorial
k × (k − 1) × … × 1. Scales each term so its derivatives match f's.
$f(x)$
The function
The curve being approximated.
$n$
Degree
How many terms you keep. More terms hug the curve over a wider range.

Match the function's value, slope, curvature and higher derivatives at one point $a$ .
For $\sin x$ at $0$ : $\;x - \tfrac{x^3}{3!} + \tfrac{x^5}{5!} - \cdots$
The error is tiny near $a$ and grows far from it.

Learn the math

Linear ApproximationsCalc I · Lesson 36, Applications of Derivatives Alternating SeriesCalc II · Lesson 31, Sequences and Series Power SeriesCalc II · Lesson 37, Power Series and Taylor Series Taylor SeriesCalc II · Lesson 39, Power Series and Taylor Series