Taylor Series

Calculus
definition

Also known as: Taylor expansion, Maclaurin series, Taylor polynomial

Grade 9-12

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A representation of a function as an infinite sum of terms calculated from the function's derivatives at a single point: f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n! Taylor series are how calculators compute \sin, \cos, e^x, and \ln.

Definition

A representation of a function as an infinite sum of terms calculated from the function's derivatives at a single point: f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n
When a = 0, it's called a Maclaurin series.

💡 Intuition

Approximate any smooth function with a polynomial by matching the function's value, slope, curvature, and all higher derivatives at a single point. The more terms you include, the wider the region where the polynomial closely matches the function. It's like fitting a polynomial glove onto the function's hand.

🎯 Core Idea

Taylor series convert transcendental functions (like e^x, \sin x, \ln(1+x)) into polynomials. The key insight is that a function is completely determined by its derivatives at a single point (within the radius of convergence).

Example

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots
e^1 \approx 1 + 1 + 0.5 + 0.167 + 0.042 + \cdots = 2.718\ldots
Each term improves the approximation.

Formula

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots

Notation

T_n(x) = Taylor polynomial of degree n. R_n(x) = remainder (error) term.

🌟 Why It Matters

Taylor series are how calculators compute \sin, \cos, e^x, and \ln. They are used in physics for linearization and perturbation theory, in numerical methods for approximation algorithms, and in analysis for proving theorems about functions.

💭 Hint When Stuck

Build a table of n, f^(n)(a), and f^(n)(a)/n! for the first few terms to spot the pattern in the coefficients.

Formal View

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n for |x - a| < R (radius of convergence). Taylor's theorem with remainder: f(x) = T_n(x) + R_n(x) where R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} for some c between a and x (Lagrange form).

🚧 Common Stuck Point

A Taylor series may only equal the function within a certain radius of convergence. Outside this radius, the series diverges and is useless. For example, \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots only converges for -1 < x \leq 1.

⚠️ Common Mistakes

  • Forgetting the n! in the denominator: the coefficient of (x-a)^n is \frac{f^{(n)}(a)}{n!}, not f^{(n)}(a).
  • Confusing Taylor polynomials (finite, approximate) with Taylor series (infinite, exact within radius of convergence): 1 + x + x^2 approximates \frac{1}{1-x}, but equals it only as 1 + x + x^2 + x^3 + \cdots.
  • Assuming the Taylor series always converges to the function: there exist functions (like f(x) = e^{-1/x^2} for x \neq 0, f(0) = 0) whose Taylor series converges everywhere but equals the function only at the center point.

Frequently Asked Questions

What is Taylor Series in Math?

A representation of a function as an infinite sum of terms calculated from the function's derivatives at a single point: f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n
When a = 0, it's called a Maclaurin series.

Why is Taylor Series important?

Taylor series are how calculators compute \sin, \cos, e^x, and \ln. They are used in physics for linearization and perturbation theory, in numerical methods for approximation algorithms, and in analysis for proving theorems about functions.

What do students usually get wrong about Taylor Series?

A Taylor series may only equal the function within a certain radius of convergence. Outside this radius, the series diverges and is useless. For example, \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots only converges for -1 < x \leq 1.

What should I learn before Taylor Series?

Before studying Taylor Series, you should understand: derivative, differentiation rules, infinite geometric series, convergence divergence.

Next Steps

power series

How Taylor Series Connects to Other Ideas

To understand taylor series, you should first be comfortable with derivative, differentiation rules, infinite geometric series and convergence divergence. Once you have a solid grasp of taylor series, you can move on to power series.

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