# Taylor Series

## Single Variable Taylor Series

If we know a function value and its derivatives at some point
`x`_{0}, we can use a *Taylor series* to
approximate its value at a nearb point `x` as

If `x-x`_{0} is small then we can truncate the series
to find

Note that for an `n`^{th} order polynomial functions,
the Taylor series converges in exactly `n`+1 terms.

Taylor series approximation of various functions showing the effect of retaining up to 4 terms in the expansion. The expansions are: Note that the third expansion, recovers the function exactly at all points since all higher derivatives (and therefore all higher terms in the expansion) are zero.

The expansions are: Note that as we retain more terms we become increasingly more accurate. However, this is an infinite series for the exponential function since all of its derivatives are nonzero.

## Multivariate Taylor Series

Consider the case where we have several equations that are functions of multiple variables,

We can write a multivariate Taylor series expansion for the
`i`^{th} function as

where is the location where we evaluate the function and is the number of independent variables.

The term is a
matrix called the **Jacobian** matrix, It
is used in solving nonlinear systems of equations using
Newton's method.

### Example

Consider the equations

Here . Let's first look at the expansion for . Applying our general equation above we find

The partial derivatives of are:

0 0 0 0

substituting this we find

We can do the same thing for

The partial derivatives of are:

-6 0 0 0 0

substituting this we find