The Taylor Series

The Taylor series is one of the most powerful and elegant tools in calculus and mathematical analysis. In simple terms, it's a way to represent a wide variety of functions as infinite sums of polynomials, built from the function's derivatives at a single point.

Core Intuition

Imagine you have a complicated, curvy function (like sin(x), eˣ, or ln(x)). A Taylor series asks: "What if we could approximate this complex curve, near a specific point, using a simple polynomial?" And not just approximate, but do it perfectly if we use an infinitely long polynomial.

The Big Idea: Build the "Best" Polynomial Match

The goal is to create a polynomial P(x) that behaves exactly like your function f(x) at and around a chosen point x = a.

How do we ensure a good match? We make sure that at the point x = a:

• The values are equal: P(a) = f(a)
• The slopes (first derivatives) are equal: P'(a) = f'(a)
• The curvatures (second derivatives) are equal: P''(a) = f''(a)
• And so on, for all higher-order derivatives.

By matching all derivatives at that point, the polynomial and the function become indistinguishable near x = a.

The Formula

The Taylor series of a function f(x) about the point x = a is:

f(x) = f(a) + f'(a)(x-a) + f''(a)/2! (x-a)² + f'''(a)/3! (x-a)³ + ...

Or, written compactly:

f(x) = Σ [n=0 to ∞] [ f⁽ⁿ⁾(a) / n! ] * (x - a)ⁿ

Where:

• f⁽ⁿ⁾(a) is the n-th derivative of f evaluated at x = a.
• n! is the factorial of n (e.g., 3! = 6).
• (x - a)ⁿ is the difference from the center point, raised to the n-th power.

Special Case: Maclaurin Series

When the center point is a = 0, the series has a simpler form and is called a Maclaurin series:

f(x) = f(0) + f'(0)x + f''(0)/2! x² + f'''(0)/3! x³ + ...

A Classic Example: eˣ

The exponential function is perfect for demonstration. Its derivative is always eˣ.

Center at a = 0 (Maclaurin Series):

• f(0) = e⁰ = 1
• f'(0) = e⁰ = 1
• f''(0) = e⁰ = 1 ... Every derivative at 0 is 1.

Plugging into the formula:

eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + ...

This infinite polynomial equals eˣ for all real numbers x.

Why is it So Important?

Approximation

By taking only the first few terms, you get a great polynomial approximation.

• eˣ ≈ 1 + x (good for very small x)
• eˣ ≈ 1 + x + x²/2 (better)

These are used constantly in engineering, physics, and computer calculations.

Understanding Functions

It reveals the "DNA" of a function in terms of its derivatives at a point.

Solving Problems

Many difficult problems (especially differential equations or integrals) become easier when you replace a function with its Taylor series.

Extending Functions

It allows us to define functions for complex numbers (like e^(iθ)) and in higher dimensions.

Connection to Calculus

It provides a deep theoretical foundation, linking differentiation and integration to polynomial behavior.

Crucial Caveats: When Does It Work?

Radius of Convergence

A Taylor series may only equal the original function within a certain distance from the center point a. For example, the series for 1/(1-x) converges only for |x| < 1.

Analytic Functions

A function that is equal to its Taylor series in a neighborhood of a point is called analytic at that point. Most common functions (sine, cosine, exponential) are analytic everywhere.

The Infamous Counterexample

The function f(x) = e^(-1/x²) (with f(0)=0) has all derivatives equal to 0 at x=0. Its Taylor series is just 0, which does not equal the function (except at 0). This shows a function can be infinitely differentiable but not analytic.

Summary

The Taylor series is the representation of a function as an infinite sum of polynomial terms, calculated from the function's derivatives at a single point. It's a bridge between the simple world of polynomials and the complex world of transcendental functions, serving as a fundamental tool for approximation, computation, and theoretical analysis in mathematics and science.