Taylor Series Formula
The Taylor series expands a function as an infinite polynomial around any point.
Learn the general formula with practical examples.
The Formula
The Taylor series represents a function as an infinite sum of terms, each involving a derivative of the function evaluated at a chosen center point a. When a = 0, the Taylor series becomes the Maclaurin series.
This series is named after English mathematician Brook Taylor, who published the concept in 1715. The Taylor series is one of the most important tools in calculus and mathematical analysis. It allows complex functions to be approximated by polynomials, which are much easier to compute.
The accuracy of the approximation depends on how many terms you include and how close x is to the center point a. The remainder term Rₙ(x) measures the error after using n terms. Not all functions have convergent Taylor series everywhere — the radius of convergence determines the range of valid approximations.
Variables
| Symbol | Meaning |
|---|---|
| f(x) | The function being approximated |
| a | The center point of the expansion |
| f⁽ⁿ⁾(a) | The n-th derivative of f evaluated at x = a |
| n! | Factorial of n (0! = 1) |
| (x - a)ⁿ | The displacement from center, raised to the n-th power |
Example 1
Find the first 3 terms of the Taylor series for ln(x) centered at a = 1.
f(x) = ln(x), so f(1) = 0
f'(x) = 1/x, so f'(1) = 1
f''(x) = -1/x², so f''(1) = -1
Taylor series: 0 + 1·(x-1) + (-1/2!)(x-1)² = (x-1) - (x-1)²/2
ln(x) ≈ (x-1) - (x-1)²/2 near x = 1
Example 2
Use the Taylor series for √x centered at a = 4 to approximate √4.1.
f(x) = √x = x^(1/2). f(4) = 2
f'(x) = 1/(2√x). f'(4) = 1/4 = 0.25
f''(x) = -1/(4x^(3/2)). f''(4) = -1/32 = -0.03125
√4.1 ≈ 2 + 0.25(0.1) + (-0.03125/2)(0.1)² = 2 + 0.025 - 0.000156
√4.1 ≈ 2.02484 (actual: 2.02485)
When to Use It
Taylor series are indispensable in advanced mathematics and applied sciences.
- Approximating difficult functions with simple polynomials
- Deriving error bounds for numerical methods
- Solving differential equations via power series
- Physics and engineering linearization of nonlinear systems