Taylor Series Calculator

A Taylor series represents a function as an infinite sum of terms, each involving a power of (x - a) multiplied by the function's nth derivative at a, divided by n!. A Maclaurin series is a Taylor series centred at a = 0. For example, e^x = 1 + x + x^2/2! + x^3/3! + ... for all x.

Taylor series allow complex functions to be approximated by polynomials, which are easy to compute and integrate. They reveal the local behaviour of functions near a point. They are used in numerical methods, physics, engineering, and computer algorithms. For example, calculators often compute sin(x) using a truncated Taylor series.

The error using n terms is bounded by the Lagrange remainder: |f(x) - Pn(x)| is at most |x|^(n+1) * M / (n+1)!, where M is the maximum of |f^(n+1)| on the interval. For alternating series, the error is bounded by the absolute value of the first omitted term. More terms always give a better approximation.

The Taylor series converges to the function only within a certain interval called the radius of convergence. For e^x and sin(x) and cos(x), the radius is infinite (they converge everywhere). For ln(1+x), convergence requires -1 less than x at most 1. For 1/(1-x), convergence requires |x| less than 1. For arctan(x), convergence requires |x| at most 1.

A Taylor series expresses a function as a sum of power terms (polynomials) centred at a point, and is useful for functions that are smooth near that point. A Fourier series expresses a periodic function as a sum of sine and cosine terms, and is useful for periodic or piecewise functions. Both are types of series expansions but serve different purposes.