STROUD Worked examples and exercises are in the text PROGRAMME 14 SERIES 2

STROUD

Worked examples and exercises are in the text

PROGRAMME 14

SERIES 2

STROUD


Power series

Standard series

The binomial series

Approximate values

Limiting values – indeterminate forms

L’Hôpital’s rule for finding limiting values

Taylor’s series

Programme 14: Series 2

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Power series

Standard series

The binomial series

Approximate values



Taylor’s series


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Power series

Introduction

Maclaurin’s series


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Power series

Introduction


When a calculator evaluates the sine of an angle it does not look up the value in a table. Instead, it works out the value by evaluating a sufficient number of the terms in the power series expansion of the sine. The power series expansion of the sine is:

This is an identity because the power series is an alternative way of way of describing the sine. The words ad inf (ad infinitum) mean that the series continues without end.

3 5 7 9 11

sin ad inf3! 5! 7! 9! 11!

x x x x xx x

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Power series

Introduction


What is remarkable here is that such an expression as the sine of an angle can be represented as a polynomial in this way.

It should be noted here that x must be measured in radians and that the expansion is valid for all finite values of x – by which is meant that the right-hand converges for all finite values of x.

3 5 7 9 11

sin ad inf3! 5! 7! 9! 11!

x x x x xx x

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Power series

Maclaurin’s series


If a given expression f (x) can be differentiated an arbitrary number of times then provided the expression and its derivatives are defined when x = 0 the expression it can be represented as a polynomial (power series) in the form:

This is known as Maclaurin’s series.

2 3 4(0) (0) (0)( ) (0) (0) ad inf

2! 3! 4!

ivf f ff x f xf x x x

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Power series

Standard series

The binomial series

Approximate values



Taylor’s series


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Standard series


The Maclaurin series for commonly encountered expressions are:

Circular trigonometric expressions:

3 5 7

sin3! 5! 7!

x x xx x

2 4 6

cos 12! 4! 6!

x x xx

3 52tan

3 15

x xx x valid for −/2 < x < /2

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Standard series


Hyperbolic trigonometric expressions:

3 5 7

sinh3! 5! 7!

x x xx x

2 4 6

cosh 12! 4! 6!

x x xx

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Standard series


Logarithmic and exponential expressions:

2 3 4 5

ln(1 )2 3 4 5

x x x xx x

2 3 4 5

12! 3! 4! 5!

x x x x xe x

valid for −1 < x < 1

valid for all finite x

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Power series

Standard series

The binomial series

Approximate values



Taylor’s series


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The binomial series


The same method can be applied to obtain the binomial expansion:

2 3

(1 ) 1 ( 1) ( 1)( 2) 2! 3!

valid for 1 1

n x xx nx n n n n n

x

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Power series

Standard series

The binomial series

Approximate values



Taylor’s series


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Approximate values


1 2 32

1 (0.02) 1 1 (0.02) 1 1 3(1 0.02) 1 0.02

2 2! 2 2 3! 2 2 2

1 11 0.01 (0.0004) (0.000008)

8 161 0.01 0.00005 0.0000005

1.0100005 0.000050

1.0099505 and so 1.

02 1.00995 to 5 dp

The Maclaurin series expansions can be used to find approximate numerical values of expressions. For example, to evaluate correct to 5 decimal places:

1.02

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Power series

Standard series

The binomial series

Approximate values



Taylor’s series


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Power series expansions can sometimes be employed to evaluate the limits of indeterminate forms. For example:

3 5

3 53

3 30 0 0

2

0

2

3 15tan 2/

3 15

1 2/1

3 15

1

3

x x x

x

x xx x

x x x xLim Lim Lim x

x x

xLim

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Power series

Standard series

The binomial series

Approximate values



Taylor’s series


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To determine the limiting value of the indeterminate form:

Then, provided the derivatives of f and g exist:

( )( ) at where ( ) ( ) 0

( )

f xF x x a f a g a

g x

( ) ( )

( ) ( )x a x a

f x f xLim Lim

g x g x

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Power series

Standard series

The binomial series

Approximate values



Taylor’s series


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Taylor’s series


Maclaurin’s series:

gives the expansion of f (x) about the point x = 0. To expand about the point x = a, Taylor’s series is employed:

2 3 4(0) (0) (0)( ) (0) (0) ad inf

2! 3! 4!

ivf f ff x f xf x x x

2 3( ) ( )( ) ( ) ( ) ad inf

2! 3!

f a f af x a f a xf a x x

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Learning outcomes

Derive the power series for sin x

Use Maclaurin’s series to derive series of common functions

Use Maclaurin’s series to derive the binomial series

Derive power series expansions of miscellaneous functions using known expansions of common functions

Use power series expansions in numerical approximations

Use l’Hôpital’s rule to evaluate limits of indeterminate forms

Extend Maclaurin’s series to Taylor’s series


Documents

STROUD Worked examples and exercises are in the text PROGRAMME 14 SERIES 2