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STROUD
Worked examples and exercises are in the text
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
STROUD
Worked examples and exercises are in the text
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
STROUD
Worked examples and exercises are in the text
Power series
Introduction
Maclaurin’s series
Programme 14: Series 2
STROUD
Worked examples and exercises are in the text
Power series
Introduction
Programme 14: Series 2
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
STROUD
Worked examples and exercises are in the text
Power series
Introduction
Programme 14: Series 2
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
STROUD
Worked examples and exercises are in the text
Power series
Maclaurin’s series
Programme 14: Series 2
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
STROUD
Worked examples and exercises are in the text
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
STROUD
Worked examples and exercises are in the text
Standard series
Programme 14: Series 2
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
STROUD
Worked examples and exercises are in the text
Standard series
Programme 14: Series 2
Hyperbolic trigonometric expressions:
3 5 7
sinh3! 5! 7!
x x xx x
2 4 6
cosh 12! 4! 6!
x x xx
STROUD
Worked examples and exercises are in the text
Standard series
Programme 14: Series 2
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
STROUD
Worked examples and exercises are in the text
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
STROUD
Worked examples and exercises are in the text
The binomial series
Programme 14: Series 2
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
STROUD
Worked examples and exercises are in the text
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
STROUD
Worked examples and exercises are in the text
Approximate values
Programme 14: Series 2
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
STROUD
Worked examples and exercises are in the text
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
STROUD
Worked examples and exercises are in the text
Limiting values – indeterminate forms
Programme 14: Series 2
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
STROUD
Worked examples and exercises are in the text
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
STROUD
Worked examples and exercises are in the text
L’Hôpital’s rule for finding limiting values
Programme 14: Series 2
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
STROUD
Worked examples and exercises are in the text
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
STROUD
Worked examples and exercises are in the text
Taylor’s series
Programme 14: Series 2
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
STROUD
Worked examples and exercises are in the text
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
Programme 14: Series 2