Power series

Taylor series

A Taylor series can be viewed in two ways:

• As a way of approximating any analytic function as a polynomial
• A one-to-one correspondence between every analytic function and an order polynomial (which may be conceived as a vector space).

An order- Taylor polynomial is constructed so that its value and all its derivatives up to order match that of the original function at a point . It is determined to be

In the case of , it is called a Maclaurin polynomial

As a correspondance, we have the statement

Error

The error of an order Taylor polynomial is given by Taylor's theorem, which converges to zero as if and only if is an analytic function.

Complex functions

The notion of the Taylor series applies to complex functions as well. For any function that is analytic for some domain where ,

This series converges for any disc centred at contained by .¹ In Complex analysis MOC, a similar power series called the Laurent series includes negative powers, that is ranges through all of . This generalises the Taylor series to cover some non-analytic functions.

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