PROGRAMME F9

1. PROGRAMME F9 BINOMIAL SERIES

2. Programme F9: Binomial series Factorials and combinations Binomial series The sigma notation The exponential number e

3. Programme F9: Binomial series Factorials and combinations Binomial series The sigma notation The exponential number e

4. Programme F9: Binomial series Factorials and combinations Factorials Combinations Three properties of combinatorial coefficients

5. Programme F9: Binomial series Factorials and combinations Factorials If n is a natural number then the product of the successive natural numbers: 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

6. Programme F9: Binomial series Factorials and combinations Combinations There are different ways of arranging r different items in n different locations. If the items are identical there are r! different ways of placing the identical items within one arrangement without making a new arrangement. So, there are different ways of arranging r identical items in n different locations. This denoted by the combinatorial coefficient

7. Programme F9: Binomial series Factorials and combinations Three properties of combinatorial coefficients

8. Programme F9: Binomial series Factorials and combinations Binomial series The sigma notation The exponential number e

9. Programme F9: Binomial series Binomial series Pascal’s triangle Binomial expansions The general term of the binomial expansion

10. 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:

11. 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:

12. 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:

13. 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:

14. Programme F9: Binomial series Binomial series Binomial expansions The power 4 expansion can be written as: or as:

15. Programme F9: Binomial series Binomial series Binomial expansions The general power n expansion can be written as: This can be simplified to:

16. Programme F9: Binomial series Binomial series The general term of the binomial expansion The (r + 1)th term in the expansion of is given as:

17. Programme F9: Binomial series Factorials and combinations Binomial series The sigma notation The exponential number e

18. Programme F9: Binomial series The sigma notation General terms The sum of the first n natural numbers Rules for manipulating sums

19. 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: 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.

20. Programme F9: Binomial series The sigma notation General terms The sigma notation form of the binomial expansion is.

21. 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:

22. Programme F9: Binomial series The sigma notation Rules for manipulating sums Rule 1: Constants can be factored out of the sum Rule 2: The sum of sums

23. Programme F9: Binomial series Factorials and combinations Binomial series The sigma notation The exponential number e

24. Programme F9: Binomial series The exponential number e The binomial expansion of is given as:

25. Programme F9: Binomial series The exponential number e The larger n becomes the smaller becomes – the closer its value becomes to 0. This fact is written as the limit ofasis 0. Or, symbolically

26. Programme F9: Binomial series The exponential number e Applying this to the binomial expansion of gives:

27. Programme F9: Binomial series The exponential number e It can be shown that is a finite number whose decimal form is: 2.7182818 . . . This number, the exponential number, is denoted by e.

28. 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:

29. 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