# Taylor series

## Contents

## Definition

### About a general point

Suppose is a function that is infinitely differentiable at a point in its domain. The **Taylor series** of about is the power series given as follows:

Here's a version with the first few terms written explicitly:

### About the point 0

In the special case of the above definition where (and in particular is infinitely differentiable at 0), the Taylor series is as follows:

Here's a version with the first few terms written explicitly:

## Well defined on germs of a functions

The Taylor series operator about a point can be thought of as a mapping:

(Germs of -functions defined about ) (Formal power series centered at )

In fact, this mapping is a -algebra homomorphism that commutes with the differential structure.

Here, two functions and are said to have the same *germ* about a point if there is an open interval containing such that .

## Relation with Taylor polynomials

The Taylor series can be viewed as a limit of Taylor polynomials. The Taylor polynomial for a function at a point in the domain is the truncation of the Taylor series to powers up to the power. If we denote the polynomial by , it is given as:

Note that this is a polynomial of degree *at most* . The degree is exactly if and only if .

Whether or not the Taylor series of a function converges to the function is determined by whether or not the sequence of Taylor polynomials of the function converges to the function.

## Computation of Taylor series

To calculate the Taylor series of a function at a point , we use the following procedure:

- Compute formal expressions for , i.e., and all its derivatives, at a generic point.
- Evaluate all these at .
- Plug the result into the Taylor series formula.

### Example of exponential function

`For further information, refer: Exponential function#Taylor series`

Consider the exponential function:

i.e., .

We want to compute the Taylor series of this function at 0.

Applying the procedure above, we get:

- Formal expressions for : These are . The sequence is a constant sequence of functions with all its members equal to .
- Evaluate at 0: Since , we get . The sequence is a constant sequence with value 1 in all places.
- Plug the result into the Taylor series formula: We get:

Taylor series for is

Note that it is *also* true that the Taylor series for the exponential function converges to the exponential function everywhere; this is because the function is globally analytic. However, this fact is not *a priori* obvious, and we are *not* asserting it as part of the computation of the Taylor series.

### Example of cosine function

`For further information, refer: Cosine function#Taylor series`

Consider the cosine function:

We want to compute the Taylor series of this function at 0.

Applying the procedure above, we get:

- Formal expressions for : These are . The sequence is periodic with period 4.
- Evaluate at 0: Since , the values are . The sequence is periodic with period 4.
- Plug the result into the Taylor series formula: We get:

Taylor series for is which can be rewritten compactly as . Note that the here is *half* the exponent on , so this is a little different from the usual way of writing Taylor series.

Note that it is *also* true that the Taylor series for the cosine function converges to the cosine function everywhere; this is because the function is globally analytic. However, this fact is not *a priori* obvious, and we are *not* asserting it as part of the computation of the Taylor series.

### Example of sine function

`For further information, refer: Sine function#Computation of Taylor series`

Consider the sine function:

We want to compute the Taylor series of this function at 0.

Applying the procedure above, we get:

- Formal expressions for : These are . The sequence is periodic with period 4.
- Evaluate at 0: Since , the values are . The sequence is periodic with period 4.
- Plug the result into the Taylor series formula: We get:

Taylor series for is which can be rewritten compactly as . Note that the here is *roughly half* the exponent on , so this is a little different from the usual way of writing Taylor series.

Note that it is *also* true that the Taylor series for the sine function converges to the sine function everywhere; this is because the function is globally analytic. However, this fact is not *a priori* obvious, and we are *not* asserting it as part of the computation of the Taylor series.

## Facts

### Preservation of structure

Together, the first three facts show that the Taylor series operator is a homomorphism of -algebras that commutes with the differential structure. The fourth fact show that it preserves an additional structure:

- Taylor series operator is linear
- Taylor series operator commutes with differentiation
- Taylor series operator is multiplicative
- Taylor series operator commutes with composition

### Convergence to the original function

- Composite of Taylor series operator and power series summation operator is identity map: What this essentially says is that
*if*a power series centered at converges to on an open interval centered at , then the power series*must*equal the Taylor series of . - Composite of power series summation operator and Taylor series operator is not identity map: It is possible to have an everywhere infinitely differentiable function and a point in the domain such that the sum of the Taylor series of at is not equal to on any interval of positive radius centered at . In fact, we can arrange our example so that the power series sum agrees with
*only*at . Note that if this happens, then there cannot be any other power series centered at that converges to on a positive radius of convergence.

A function whose Taylor series at a point converges to the function in an open interval centered at the point is termed a locally analytic function at the point. If the Taylor series converges to the function everywhere, the function is termed a globally analytic function. We have that locally analytic not implies globally analytic.