1. STROUD Worked examples and exercises are in the text PROGRAMME F9 BINOMIAL SERIES

2. STROUD Worked examples and exercises are in the text Factorials and combinations Binomial series The sigma notation The exponential number e Programme F9: Binomial series

3. STROUD Worked examples and exercises are in the text Factorials and combinations Binomial series The sigma notation The exponential number e Programme F9: Binomial series

4. STROUD Worked examples and exercises are in the text Factorials and combinations Factorials Combinations Three properties of combinatorial coefficients Programme F9: Binomial series

5. STROUD Worked examples and exercises are in the text Factorials and combinations Factorials Programme F9: Binomial series 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. STROUD Worked examples and exercises are in the text Factorials and combinations Combinations Programme F9: Binomial series 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. STROUD Worked examples and exercises are in the text Factorials and combinations Three properties of combinatorial coefficients Programme F9: Binomial series

8. STROUD Worked examples and exercises are in the text Factorials and combinations Binomial series The sigma notation The exponential number e Programme F9: Binomial series

9. STROUD Worked examples and exercises are in the text Binomial series Pascal’s triangle Binomial expansions The general term of the binomial expansion Programme F9: Binomial series

10. STROUD Worked examples and exercises are in the text Binomial series Pascal’s triangle Programme F9: Binomial series 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. STROUD Worked examples and exercises are in the text Binomial series Pascal’s triangle Programme F9: Binomial series Evaluating the combinatorial coefficients gives a triangular array of numbers that is called Pascal’s triangle:

12. STROUD Worked examples and exercises are in the text Binomial series Binomial expansions Programme F9: Binomial series 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. STROUD Worked examples and exercises are in the text Binomial series Binomial expansions Programme F9: Binomial series Notice that the coefficients in the expansions are the same as the numbers in Pascal’s triangle:

14. STROUD Worked examples and exercises are in the text Binomial series Binomial expansions Programme F9: Binomial series The power 4 expansion can be written as: or as:

15. STROUD Worked examples and exercises are in the text Binomial series Binomial expansions Programme F9: Binomial series The general power n expansion can be written as: This can be simplified to:

16. STROUD Worked examples and exercises are in the text Binomial series The general term of the binomial expansion Programme F9: Binomial series The (r + 1)th term in the expansion of is given as:

17. STROUD Worked examples and exercises are in the text Factorials and combinations Binomial series The sigma notation The exponential number e Programme F9: Binomial series

18. STROUD Worked examples and exercises are in the text The sigma notation General terms The sum of the first n natural numbers Rules for manipulating sums Programme F9: Binomial series

19. STROUD Worked examples and exercises are in the text The sigma notation General terms Programme F9: Binomial series 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. STROUD Worked examples and exercises are in the text The sigma notation General terms Programme F9: Binomial series The sigma notation form of the binomial expansion is.

21. STROUD Worked examples and exercises are in the text The sigma notation The sum of the first n natural numbers Programme F9: Binomial series The sum of the first n natural numbers can be written as:

22. STROUD Worked examples and exercises are in the text The sigma notation Rules for manipulating sums Programme F9: Binomial series Rule 1: Constants can be factored out of the sum Rule 2: The sum of sums

23. STROUD Worked examples and exercises are in the text Factorials and combinations Binomial series The sigma notation The exponential number e Programme F9: Binomial series

24. STROUD Worked examples and exercises are in the text The exponential number e Programme F9: Binomial series The binomial expansion of is given as:

25. STROUD Worked examples and exercises are in the text The exponential number e Programme F9: Binomial series The larger n becomes the smaller becomes – the closer its value becomes to 0. This fact is written as the limit of as is 0. Or, symbolically

26. STROUD Worked examples and exercises are in the text The exponential number e Programme F9: Binomial series Applying this to the binomial expansion of gives:

27. STROUD Worked examples and exercises are in the text The exponential number e Programme F9: Binomial series 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. STROUD Worked examples and exercises are in the text The exponential number e Programme F9: Binomial series 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. 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 e x where e is the exponential number