Taylor and Maclaurin Series

A Taylor series represents a function as an infinite sum of terms calculated from the function’s derivatives at a single point. It is one of the most powerful ideas in mathematics: even complicated functions like sin(x) or e^x can be approximated by simple polynomials.

The Formula

The Taylor series of f(x) around point a is: f(a) + f′(a)(x-a) + f′′(a)(x-a)²/2! + f′′′(a)(x-a)³/3! + ... When a=0, it is called a Maclaurin series.

Common Maclaurin Series

e^x = 1 + x + x²/2! + x³/3! + ... (for all x)

sin(x) = x - x³/3! + x⁵/5! - ... (for all x)

cos(x) = 1 - x²/2! + x⁴/4! - ... (for all x)

ln(1+x) = x - x²/2 + x³/3 - ... (for -1 < x ≤ 1)

Applications

Taylor series are used in numerical methods, physics simulations, engineering approximations, and even in how calculators compute sin, cos, and exponential functions internally. The more terms you include, the better the approximation.

Explore series expansions with our Taylor Series Visualizer.