Taylor series

Univariate Taylor series

Consider a function, i.e., a univariate function of a real variable, and any point in the domain of. Under certain conditions, the following is always true near:

where is the n-th derivative

This formula is known as the Taylor expansion of about the point and the right hand side is a Taylor series. This series is always convergent near (the Maclaurin series)

Conditions

For the Taylor series of a function to exist at a point, the following conditions must be satisfied:

1. The function must be infinitely differentiable at the point . This means that all derivatives for must exist.
   1. This condition is relaxed when we only need some -order Taylor polynomial (where is small): the function must be sufficiently differentiable – it just needs to have enough continuous derivatives () to satisfy the requirements
2. The Taylor series must converge to for around . That is, the remainder term in the Taylor expansion must approach zero as , ensuring the series represents the function locally around

Multivariate Taylor series

For a function (an variable function over real numbers) and a point vector in the domain of , the general Taylor series expansion is:

where:

• is the gradient of at
• is the Hessian matrix (the matrix of second-order partial derivatives) of at .

Usage

• approximate an infinitely differentiable function at around
• estimate the value of the derivative of a sufficiently differentiable function