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Taylor Series Expansion

Approximate complex functions using polynomials with this fundamental tool in mathematical computing.

📘 Start Series Expansion

What is a Taylor Series?

A Taylor series represents a function as an infinite sum of polynomial terms centered at $ a $. The series is named after Brook Taylor.

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

Examples

  • $ e^x = 1 + \frac{x}{1!} + \frac{x²}{2!} + \frac{x³}{3!} + \ldots $
  • $ \sin(x) $ expanded around $ x=0 $: $ x - x³/6 + x⁵/120 $

The more terms you use in the series, the better the approximation near the center $ a $.

Taylor Series Formula

The general form for a function $ f(x) $ around a point $ a $ is:

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

where $ f^{(n)} $ is the nth derivative. When a=0, it becomes Maclaurin series.

Real-World Applications

Scientific Computing

Used in physics for approximations in quantum mechanics or fluid dynamics calculations.

Machine Learning

Helps approximate complex activation functions like tanh(x) and exp(x).

Try Applying Taylor Series

Series expansion: f(x) = 1 + x + x²/2 + x³/6 + x⁴/24 + x⁤/120...

Common Taylor Series Expansions

Exponential Function

$ e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + $ ...

Sine Function

$ \sin x = x - x³/3! + x⁵/5! - x⁷/7! $

Cosine Function

$ \cos x = 1 - x²/2! + x⁴/4! - x⁶/6! $

Key Concepts

Try Calculating Your Own Series

Calculate Series Expansions