Transcript 12 Series 2 PowerPoint

Programme 12: Series 2
PROGRAMME 12
SERIES 2
STROUD
Worked examples and exercises are in the text
Programme 12: Series 2
Power series
Approximate values
Limiting values – indeterminate forms
STROUD
Worked examples and exercises are in the text
Programme 12: Series 2
Power series
Approximate values
Limiting values – indeterminate forms
STROUD
Worked examples and exercises are in the text
Programme 12: Series 2
Power series
Introduction
Maclaurin’s series
Standard series
The binomial series
STROUD
Worked examples and exercises are in the text
Programme 12: Series 2
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:
x 3 x5 x 7 x 9 x11
sin x º x - + - + +… ad inf
3! 5! 7! 9! 11!
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.
STROUD
Worked examples and exercises are in the text
Programme 12: Series 2
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.
x 3 x5 x 7 x 9 x11
sin x º x - + - + +… ad inf
3! 5! 7! 9! 11!
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.
STROUD
Worked examples and exercises are in the text
Programme 12: Series 2
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:
iv
f ¢¢(0)
3 f ¢¢¢(0)
4 f (0)
f (x) º f (0) + xf ¢(0) + x
+x
+x
+… ad inf
2!
3!
4!
2
This is known as Maclaurin’s series.
STROUD
Worked examples and exercises are in the text
Programme 12: Series 2
Power series
Standard series
The Maclaurin series for commonly encountered expressions are:
Circular trigonometric expressions:
x 3 x5 x 7
sin x º x - + - +…
3! 5! 7!
x2 x4 x6
cos x º 1- + - +…
2! 4! 6!
x 3 2x 5
tan x º x + +
+…
3 15
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valid for −θ/2 < x < θ/2
Worked examples and exercises are in the text
Programme 12: Series 2
Power series
Standard series
Hyperbolic trigonometric expressions:
x 3 x5 x 7
sinh x º x + + + +…
3! 5! 7!
x2 x4 x6
cosh x º 1+ + + +…
2! 4! 6!
STROUD
Worked examples and exercises are in the text
Programme 12: Series 2
Power series
Standard series
Logarithmic and exponential expressions:
x 2 x 3 x 4 x5
ln(1+ x) º x - + - + -…
2
3 4 5
x 2 x3 x 4 x5
e º 1+ x + + + + +…
2! 3! 4! 5!
x
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valid for −1 < x < 1
valid for all finite x
Worked examples and exercises are in the text
Programme 12: Series 2
Power series
The binomial series
The same method can be applied to obtain the binomial expansion:
x2
x3
(1+ x) º 1+ nx + n(n - 1) + n(n - 1)(n - 2) +…
2!
3!
n
valid for - 1 < x < 1
STROUD
Worked examples and exercises are in the text
Programme 12: Series 2
Power series
Approximate values
Limiting values – indeterminate forms
STROUD
Worked examples and exercises are in the text
Programme 12: Series 2
Power series
Approximate values
Limiting values – indeterminate forms
STROUD
Worked examples and exercises are in the text
Programme 12: Series 2
Approximate values
The Maclaurin series expansions can be used to find approximate
numerical values of expressions. For example, to evaluate 1.02 correct
to 5 decimal places:
1
2
3
æ
ö
1
(0.02)
1
1
(0.02)
1 æ 1ö æ 3ö
(1+ 0.02) 2 º 1+ 0.02 +
+
+
- ÷ ç - ÷ +…
ç
÷
ç
2
2! 2 è 2 ø
3! 2 è 2 ø è 2 ø
(
)
1
1
= 1+ 0.01- (0.0004) + (0.000008) -…
8
16
= 1+ 0.01- 0.00005 + 0.0000005 -…
= 1.0100005 - 0.000050
= 1.0099505
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and so 1.02 = 1.00995 to 5 dp
Worked examples and exercises are in the text
Programme 12: Series 2
Approximate values
Taylor’s series
Maclaurin’s series:
iv
f ¢¢(0)
3 f ¢¢¢(0)
4 f (0)
f (x) º f (0) + xf ¢(0) + x
+x
+x
+… ad inf
2!
3!
4!
2
gives the expansion of f (x) about the point x = 0. To expand about the
point x = a, Taylor’s series is employed:
f (x + a) º f (a) + xf ¢(a) + x 2
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f ¢¢(a)
f ¢¢¢(a)
+ x3
+… ad inf
2!
3!
Worked examples and exercises are in the text
Programme 12: Series 2
Series of powers of the natural numbers
Sum of natural numbers
The series:
n
1+ 2 + 3+ 4 + 5 +… + n = å r
r=1
is an arithmetic series with a = 1 and d = 1 so that:
n
r 
r 1
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n
n(n  1)
(2a  [n  1]d ) 
2
2
Worked examples and exercises are in the text
Programme 12: Series 2
Power series
Approximate values
Limiting values – indeterminate forms
STROUD
Worked examples and exercises are in the text
Programme 12: Series 2
Limiting values – indeterminate forms
Power series expansions can sometimes be employed to evaluate the limits
of indeterminate forms. For example:
ìæ
ü
ö
x 3 2x 5
+… ÷ - x/ ï
ï ç x/ + +
ì
ö 3ü
3
15
ì tan x - x ü
è
ø
ï
ï
ïæ x 3 2x 5
ï
Lim í
=
Lim
=
Lim
+
+…
/
x
ý
í
ý
í
ý
3
3
÷ø
x®0
x®0
x®0 ç
3
15
x
x
è
ï
ï
î
þ
ï
ï
î
þ
ï
ï
î
þ
ì
ö ü
ïæ 1 2x 2
ï
= Lim íç +
+… ÷ / 1ý
x®0 è 3
15
ø ï
ï
î
þ
1
=
3
STROUD
Worked examples and exercises are in the text
Programme 12: Series 2
Limiting values – indeterminate forms
L’Hôpital’s rule for finding limiting values
To determine the limiting value of the indeterminate form:
F ( x) 
f ( x)
at x  a where f (a)  g (a)  0
g ( x)
Then, provided the derivatives of f and g exist:
 f ( x) 
 f ( x) 
Lim 

Lim



x a
 g ( x)  xa  g ( x) 
This is known as L’Hôpital’s rule
STROUD
Worked examples and exercises are in the text
Programme 12: Series 2
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
mcommon 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
STROUD
Worked examples and exercises are in the text