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Taylor and Maclaurin Series

- Taylor Polynomials at x 0
- Basic Taylor Polynomials
- Taylor Series and Maclaurin Series
- Taylor Polynomials at x a
- Finding Taylor Series by Substitutions,

Differentiation and Integration

Functions Represented by Power Series

Consider a function f represented by a

converging power series f(x) a0 a1x a2x2

a3x3 .

Clearly f(0) a0.

To compute the other coefficients ak,

differentiate term by term to get f(x) a1

2a2x .

Insert x 0 to the above to get f(0) a1.

Conclude

A converging power series representing a function

f is necessarily of the above form. This power

series is called Maclaurin Series, a special form

a more general Taylor Series.

Approximating Functions

Another way to get to Taylor Series is to

consider approximations of functions by

polynomials.

We assume that the function f is has

derivatives of all orders everywhere in its

domain of definition.

The Taylor polynomial of

degree n for given

function f at a point a is a polynomial P

of degree n such that P(k)

(a)f (k)(a) for k0,1,,n. This means that

the value of the polynomial P and all of its

derivatives up to the order n agree with those

of the function f at the point xa.

Definition

Observe that the defining conditions for the

Taylor polynomial have to do with the behavior of

the polynomial at one point only.

Taylor Polynomials at x 0

Straightforward differentiation yields.

The general formula is

We conclude that the Taylor Polynomial for an

infinitely differentiable function f at x0 is

uniquely defined, and that the coefficients ak

are given by the above formula.

Explicit Taylor Polynomials

Formula

By the preceding considerations, the Taylor

Polynomial of degree n of a function f is the

polynomial Tf,n(x)

- Using this formula Taylor polynomials of

functions can often be rather easily computed.

Strategy is the following - Compute several derivatives of the given

function. - Evaluate these derivatives at x 0.
- Detect a pattern to find a general formula for

f(n)(0).

Taylor Polynomials for the Sine Function

To find the Taylor polynomials of the function

f(x) sin(x) compute derivatives and evaluate

them at x 0. One gets

Conclude

All even order derivatives of sin take the value

0 at x 0. Odd order derivatives take the

values 1 and -1.

Conclusion as a Formula

Formula

Taylor Polynomials for Cosine

To find the Taylor polynomials of the function

g(x) cos(x) compute derivatives and evaluate

them at x 0. One gets

Conclude

All odd order derivatives of cos take the value 0

at x 0. Even order derivatives take the

values 1 and -1.

Conclusion as a Formula

Formula

Taylor Polynomials for the Exponential Function

To find the Taylor polynomials of the exponential

function h(x) ex compute derivatives and

evaluate them at x 0. One gets

Conclude

The value of the exponential function and that of

all of its derivatives at x 0 is 1.

Formula

Taylor Polynomials for the Sine Function

Formula

The following figure illustrates Taylor

polynomials of degrees 5 (blue), 9 (red) and 15

(green) for the sine function.

One concludes from the picture that all of the

above Taylor approximations for the sine function

appear to approximate the function well near the

origin (center of the above picture). Higher

order Taylor polynomials approximate better away

from the origin.

Basic Taylor Polynomials

1

2

3

We will later see that the above polynomials can

be used to approximate the values of the

respective functions for all x.

The Taylor Polynomials of the function f(x)

(1x)p are given below. They can be used to

approximate the values of the function only for

-1ltxlt1.

Taylor Polynomials at x a

The conditions used to define a Taylor polynomial

P of a given function f require that the

polynomial P and all of its non-zero

derivatives at a point xa agree with those of

the function f. Clearly the definition implies

that the polynomial P approximates the function

f best near the point xa.

Formula for Taylor Polynomials at xa

Assume the function f has all derivatives at

the point xa.

Taylor polynomial of degree n at xa is

The above formula follows by directly computing

the values of the derivatives of the function f

and those of the polynomial P at xa.

Goodness of Approximations

The following figure shows the graph of the sine

functions and those of its Taylor polynomials of

degree 5 at the points x-p, x0, xp, x2p,

x3p.

The Taylor polynomial of degree 5 at the point

x-p approximates the sine function so well near

the point x-p that its graphs is completely

covered by the black graph of the sine function

near that point. As xlt -3p/2 or xgt -p/2 the

approximation fails to follow the graph of the

sine function. These portions of the graph of

the Taylor polynomial of degree 5 at x-p are

shown as the left most red graphs above and under

the x -axis.

Taylor Series and Maclaurin series

Letting n grow the Taylor polynomials at x a

define Taylor series at x a for the respective

functions. Basic Taylor Polynomials yield the

following Basic Taylor Series at x 0. The

Taylor Series at x 0 are also called Maclaurin

series.

These series converge and represent the given

function for all x. This will be shown later.

The Binomial Series

This is valid for -1ltxlt1.

Taylor Polynomial Approximations

Problem

Solution

The error is given by

The Taylor series for sin(x) is an alternating

series.

The error done when approximating sin(x) by a

Taylor polynomial of degree 5 is bounded by the

absolute value of the first term left out. Hence

Binomial Series and Geometric Series

The Binomial Series

Insert p -1 to the above to get

The binomial series for p -1, is the geometric

series with the first term 1 and with q x as

the ratio of two subsequent terms.

Conclude

Binomial Series Example

Problem

Find Taylor series for the function f(x) 1/(1

- x)2.

Use the Binomial Series

Insert p -2 to the above to get

Alternative derivation for this series expansion

follows later.

Finding Taylor Series

- One can find Taylor series for complicated

functions by - Substitutions
- Integrating a known series term by term
- Differentiating a known series term by term
- Any combination of the above tricks

One usually starts with one of the basic Taylor

series and manipulates that to get the desired

Taylor series. The above tricks are legal

provided that the series in question converge and

represent the functions in question. This

depends on the function for which Taylor series

representation needs to be derived. Many of the

basic Taylor series converge everywhere.

Finding Taylor Series by Substitution

Problem

Solution

Finding Taylor Series by Integration

Problem

Solution

Taylor Series for ln(1 x)

Formula

This figure shows the graph of the function ln(1

x) and those of its Taylor polynomials of order

4 (blue), 9 (red), 14 (green) and 19 (yellow).

One observes that up to x 1 higher order

Taylor polynomials give better approximations

than lower order Taylor polynomials. For x gt 1,

the situation is reversed the higher the order,

the worse the approximation. This reflects the

fact that the Taylor series for ln(1 x) does

not converge for x gt 1.

Finding Taylor Series by Differentiation

Problem

Solution

The above formula is a special case of the

binomial series and it converges for x lt 1.

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