Transcript F09 Binomial series Powerpoint

PROGRAMME F9
BINOMIAL
SERIES
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
Factorials and combinations
Binomial series
The sigma notation
The exponential number e
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
Factorials and combinations
Binomial series
The sigma notation
The exponential number e
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
Factorials and combinations
Factorials
Combinations
Three properties of combinatorial coefficients
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
Factorials and combinations
Factorials
If n is a natural number then the product of the successive natural numbers:
n (n 1) (n  2) ( )3 21
is called n-factorial and is denoted by the symbol n!
In addition 0-factorial, 0!, is defined to be equal to 1. That is, 0! = 1
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
Factorials and combinations
Combinations
n!
There are
different ways of arranging r different items in n different
locations. (n r )!
If the items are identical there are r! different ways of placing the identical items
within one arrangement without making a new arrangement.
n!
So, there are
different ways of arranging r identical items in n different
(
n

r
)!
r
!
locations.
This denoted by the combinatorial coefficient nCr 
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n!
(n  r )!r !
Worked examples and exercises are in the text
Programme F9: Binomial series
Factorials and combinations
Three properties of combinatorial coefficients
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Cn  nC0 1
(a)
n
(b)
n
(c)
n
Cnr  nCr
Cr  nCr 1  n1Cr 1
Worked examples and exercises are in the text
Programme F9: Binomial series
Factorials and combinations
Binomial series
The sigma notation
The exponential number e
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
Binomial series
Pascal’s triangle
Binomial expansions
The general term of the binomial expansion
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
Binomial series
Pascal’s triangle
The following triangular array of combinatorial coefficients can be
constructed where the superscript to the left of each coefficient indicates the
row number and the subscript to the right indicates the column number:
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
Binomial series
Pascal’s triangle
Evaluating the combinatorial coefficients gives a triangular array of numbers
that is called Pascal’s triangle:
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
Binomial series
Binomial expansions
A binomial is a pair of numbers raised to a power. For natural number
powers these can be expanded to give the appropriate binomial series:
(a  b)1  a  b
(a  b)2  a2  2ab  b2
(a  b)3  a3  3a2b  3ab2  b3
(a  b)4  a4  4a3b  6a2b2  4ab3  b4
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
Binomial series
Binomial expansions
Notice that the coefficients in the expansions are the same as the numbers in
Pascal’s triangle:
(a  b)1  a  b
(a  b)2  a2  2ab  b2
(a  b)3  a3  3a2b  3ab2  b3
(a  b)4  a4  4a3b  6a2b2  4ab3  b4
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
Binomial series
Binomial expansions
The power 4 expansion can be written as:
(a  b)4 1a4b0  4a3b1  6a2b2  4a1b3 1a0b4
or as:
(a  b)4 4 C0a4b0  4C1a3b1  4C2a2b2  4C3a1b3  4C4a0b4
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
Binomial series
Binomial expansions
The general power n expansion can be written as:
(a  b)n n C0anb0  nC1an1b1  nC2an2b2 
 nCr anrbr 
 nCna0bn
This can be simplified to:
(a  b)n  an  nan1b 
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n(n1) n2 2 n(n1)(n2) n3 3
a b 
a b 
2!
3!
 bn
Worked examples and exercises are in the text
Programme F9: Binomial series
Binomial series
The general term of the binomial expansion
The (r + 1)th term in the expansion of (a  b)n is given as:
Cr anrbr 
n
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n!
a n  r br
(n r )!r !
Worked examples and exercises are in the text
Programme F9: Binomial series
Factorials and combinations
Binomial series
The sigma notation
The exponential number e
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
The sigma notation
General terms
The sum of the first n natural numbers
Rules for manipulating sums
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
The sigma notation
General terms
If a sequence of terms are added together:
f(1) + f(2) + f(3) + . . . + f(r) + . . . + f(n)
their sum can be written in a more convenient form using the sigma
notation:
n
 f (r )
r 1
The sum of terms of the form f(r) where r ranges in value from 1 to n.
f(r) is referred to as a general term.
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
The sigma notation
General terms
The sigma notation form of the binomial expansion is.
n n
n r r
n
n r r
 Cr a b where Cr a b is the general term
r 1
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
The sigma notation
The sum of the first n natural numbers
The sum of the first n natural numbers can be written as:
1 2  3 
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 n   r  n(n 1)
2
r 1
n
Worked examples and exercises are in the text
Programme F9: Binomial series
The sigma notation
Rules for manipulating sums
Rule 1: Constants can be factored out of the sum
n
n
kf (r)  k  f (r)

r 1
r 1
Rule 2: The sum of sums
n
n
n
{ f (r)  g (r)}   f (r)   g (r)

r 1
r 1
r 1
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
Factorials and combinations
Binomial series
The sigma notation
The exponential number e
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
The exponential number e
n
 1
The binomial expansion of 1  is given as:
 n
n
2
3
 1  n(n 1)  1 
1
 1
n
(
n

1)(
n

2)
  
1 n  1 n  n   2!  n  
3!
n


11 (11/ n)  (11/ n)(1 2/ n) 
2!
3!
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 
  1 
n
 
  1 
n
n
n
Worked examples and exercises are in the text
Programme F9: Binomial series
The exponential number e
1
The larger n becomes the smaller
becomes – the closer its value
n
becomes to 0.
This fact is written as the limit of 1 as n is 0.
n
Or, symbolically
STROUD
1
Lim    0
n  n
 
Worked examples and exercises are in the text
Programme F9: Binomial series
The exponential number e
n
 1
Applying this to the binomial expansion of 1  gives:
 n
n
 1
(10) (10)(10)
Lim 1  11


3!
n 
2!
 n

1 1 1 1
   
0! 1! 2! 3!

1
r 0 r !

STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
The exponential number e

1
It can be shown that 
is a finite number whose decimal form is:
r 0 r !
2.7182818 . . .
This number, the exponential number, is denoted by e.
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
The exponential number e
It will be shown in Part II that there is a similar expansion for the
exponential number raised to a variable power x, namely:
2
3
x
x
e  1 x   
2! 3!
x

r
x
 
r!
r
x

r 0 r !
STROUD
Worked examples and exercises are in the text
Programme F9: Binomial series
Learning outcomes
Define n! and recognise that there are n! different combinations of n different items
Evaluate n! using a calculator and manipulate expressions involving factorials
Recognize that there are different combinations of r identical items in n locations
Recognize simple properties of combinatorial coefficients
Construct Pascal’s triangle
Write down the binomial expansion for natural number powers
Obtain specific terms in the binomial expansion using the general term
Use the sigma notation
Recognize and reproduce the expansion for ex where e is the exponential number
STROUD
Worked examples and exercises are in the text