Taylor series expansion

Taylor Series Expansion A Taylor series expansion is an infinite series that represents a function in a neighborhood of a given point. It is constructed by...

Taylor Series Expansion

A Taylor series expansion is an infinite series that represents a function in a neighborhood of a given point. It is constructed by taking the limit of a sequence of polynomials.

Formally,

$f(x) = \sum_{n=0}^{\infty} a_n (x - a)^n$

where:

(f(x)) is the function whose Taylor series is being expanded.
(a_n) is the coefficient of the (n^{th}) term in the Taylor series.
The series converges for all (x) in the neighborhood of (a).

Examples:

The Taylor series expansion for the function (f(x) = \frac{1}{1-x}) around (x = 0) is:

$\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$

The Taylor series expansion for the function (f(x) = \sqrt{x}) around (x = 0) is:

$\sqrt{x} = \sum_{n=0}^{\infty} \frac{(x)^n}{n!}$

Applications of Taylor Series Expansions:

Taylor series expansions have numerous applications in mathematics and physics, including:

Finding the derivatives and integrals of functions.
Approximating functions with high accuracy.
Solving differential equations.

Note: Taylor series expansions are valid only around points where the function is differentiable