Taylor Series — Mathematics Knowledge System

Overview

A Taylor series represents a function as an infinite sum of terms built from its derivatives at a single point. It's one of the most powerful ideas in analysis: a function that might be hard to compute directly (sin, eˣ, ln) can be approximated arbitrarily well by a polynomial, which computers can evaluate quickly using only addition and multiplication.

Intuition

Near a point, you can approximate a curve with its tangent line (first-order approximation — matches value and slope). You can do better by also matching curvature, using a parabola (second-order). The Taylor series is what happens if you keep going forever: match the value, the slope, the curvature, and every higher derivative, all at once. For 'nice' functions, this infinite polynomial doesn't just approximate the function near that point — it equals the function exactly.

Formal Definition

Definition

The Taylor series of f centered at a:

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

f⁽ⁿ⁾(a) is the n-th derivative of f evaluated at a

Taylor series

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n

The special case a = 0

Maclaurin series

Notation

Notation	Meaning
$f^{(n)}(a)$	The n-th derivative of f evaluated at a
$n!$	n factorial: n·(n-1)·(n-2)···2·1

Properties

Maclaurin series for eˣ

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \cdots

Example: Converges for all real x

Maclaurin series for sin(x)

\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots

Maclaurin series for 1/(1-x)

\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n

Condition: Converges only for |x| < 1 (geometric series)

Theorems

Theorem 1: Taylor's Theorem with Remainder

f(x) = \sum_{n=0}^{N} \frac{f^{(n)}(a)}{n!}(x-a)^n + R_N(x), \text{ where } R_N(x) \to 0 \text{ as } N \to \infty \text{ for many well-behaved } f

Applications

Hardware and software math libraries compute sin, cos, exp, log using truncated Taylor (or Chebyshev) polynomial approximations — a CPU can't 'just know' sin(x).

Formula Explorer

Worked Examples

Compute derivatives at 0: f(0)=1, f'(0)=0, f''(0)=-1, f'''(0)=0, f⁗(0)=1.
$\cos(0)=1,\ -\sin(0)=0,\ -\cos(0)=-1$
Assemble the series using f⁽ⁿ⁾(0)/n! xⁿ.
$1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$

Answer: 1 - x²/2 + x⁴/24 - ...

Practice Problems

Difficulty 6/10

Write the first three terms of the Taylor series of ln(1+x) centered at 0.

Difficulty 5/10

Engineers use the 'small-angle approximation' sin θ ≈ θ for small θ. Justify it with the Taylor series of sin θ, and estimate the error at θ = 0.1 rad.

Difficulty 5/10

Use the Taylor series of eˣ to approximate e^0.1 with three terms, and compare to the true value.

Common Mistakes

Common Mistake

Assuming a Taylor series always converges to the function everywhere.

It only converges within a radius of convergence, and even where it converges it doesn't always equal f(x) — a classic counterexample is f(x)=e^{-1/x²}, whose Maclaurin series is identically 0 but f itself is not.

Quiz

The Maclaurin series is a Taylor series centered at:

x = 1x = a (arbitrary)x = 0x = ∞

Flashcards

1 / 2

Historical Background

James Gregory and Isaac Newton discovered special cases of Taylor series in the 1670s. Brook Taylor published the general method in 1715 in Methodus Incrementorum Directa et Inversa, though he didn't fully explore its implications. Colin Maclaurin popularized the special case centered at zero (now called the Maclaurin series) in his 1742 Treatise of Fluxions, and it's his name attached to that special case today.

1671
Gregory discovers a series expansion technique
James Gregory
1715
Brook Taylor publishes the general theorem
Brook Taylor
1742
Maclaurin popularizes the a=0 special case
Colin Maclaurin

Summary

A Taylor series represents f(x) as an infinite polynomial built from derivatives at a point a.
Maclaurin series is the special case centered at a=0.
Key series to memorize: eˣ, sin(x), cos(x), 1/(1-x) (geometric).
Only converges within a radius of convergence, and doesn't always equal f(x) even there.
Used practically to compute transcendental functions in software and to approximate curvature in optimization.

References

BookTaylor, B. (1715). Methodus Incrementorum Directa et Inversa.
WebsiteWikipedia — Taylor series