Taylor's series

Taylor’s Series

Definition

• The Taylor’s series for a function about a point a is a way to represent that function as an infinite sum of terms based on the function’s derivatives at the given point. It’s written as f(x) = f(a) + f’(a)(x-a) + f’‘(a)(x-a)²/2! + ….

Basic Constituents

• f’(a), f’‘(a), etc. are the derivatives of the function evaluated at the point a.
• (x-a), (x-a)² etc. are the powers of (x-a) and represent the distance of x from a.
• 1!, 2! etc. are factorial terms which grow very quickly because n! = n(n-1)(n-2)…321.

Approximations Using Taylor’s Series

• By truncating the series after a finite number of terms, the Taylor series provides an approximation of the function. This approximation improves as the number of terms increases.
• For example, you can approximate a polynomial of degree n accurately with the sum of the first (n + 1) terms of its Taylor series.

Remainder Theorem

• The remainder after n terms in Taylor’s series gives the difference between the true function value and its approximation using n terms. Taylor’s Remainder Theorem provides bounds for this difference.

Interval of Convergence

• The interval of convergence is the set of all x for which the series converges to f(x).
• For some functions, the series diverges (does not sum to f(x)) for all x values outside a certain interval around a.

Maclaurin Series

• A Maclaurin series is a special case of a Taylor series where the series is expanded about a=0. It’s written as f(x) = f(0) + f’(0)x + f’‘(0)x²/2! + …
• It simplifies calculations in some circumstances, as far as the function f(x) has a Maclaurin series representation.

It’s important to understand each of these concepts and know when to use them. The Taylor series is a powerful tool in calculus, particularly in solving problems that can’t be solved through elementary methods.