### Using Lagrange theorem to approximate values

Recall that Lagrange Theorem states that for every function differentiable in an interval , there exists in this interval, such that

thus

but for small we can approximate and get the value approximated by the values of and at , so:

### Taylor’s Theorem

Taylor’s Theorem is a very powerful generalization of Lagrange Theorem. It gives us a possibility to approximate a function with polynomials.

Let and has all derivatives, up to -th. Let, then:

where is some number between and .

Sum will be called -th order Taylor polynomial of in point , and is the -th remainder term.

### Calculating approximations

Taylor’s Theorem enables calculating some approximate values of a function in a given point. E.g., . Its second order Taylor polynomial in is , which for is and it is up to the remainder term, so the possible error is not greater than .

### Taylor series

Therefore, if on a given interval , then we can write as a sum of a series. E.g. for is convergent on the whole real line, so the Taylor series of in point (it is easy because ) is:

Such a series but for is called a Maclaurin series.