A Taylor series approximates a smooth function near a chosen point aa with a polynomial that matches the function's derivatives at that point.

Explorer

Use the controls to choose a function, move the expansion point, and increase the number of terms. The graph compares the exact function to the Taylor polynomial, and the error plot shows where the approximation starts to drift.

f(x) = e^x
Maclaurin series is the special case a = 0.
Converges for every real x.
a = 0window: -3 to 3
Black: exact functionBlue: Taylor polynomialDashed: expansion point
Approximation error: polynomial minus exact value
mean absolute error: 0.8563max absolute error: 7.0855
Current polynomial
T3(x) = 1 + x + 0.5x^2 + 0.1667x^3

Formula

For a function ff, the first NN terms of the Taylor expansion about aa are:

f(x)≈∑n=0N−1f(n)(a)n!(x−a)nf(x) \approx \sum_{n=0}^{N-1} \frac{f^{(n)}(a)}{n!}(x-a)^n

When a=0a = 0, this is called a Maclaurin series.

What to Notice

  1. The polynomial is usually most accurate near x=ax = a.
  2. Adding terms usually improves the local fit.
  3. Some functions have a limited convergence radius because of nearby singularities.

Common Maclaurin Series

Function First terms
exe^x 1+x+x2/2!+x3/3!+⋯1 + x + x^2/2! + x^3/3! + \cdots
sin⁡(x)\sin(x) x−x3/3!+x5/5!−⋯x - x^3/3! + x^5/5! - \cdots
cos⁡(x)\cos(x) 1−x2/2!+x4/4!−⋯1 - x^2/2! + x^4/4! - \cdots
ln⁡(1+x)\ln(1 + x) x−x2/2+x3/3−⋯x - x^2/2 + x^3/3 - \cdots

Taylor series are a core tool in calculus because they turn complicated functions into polynomials that are easier to analyze, differentiate, and compute.

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