1. 22C:19 Discrete Structures Sequence and Sums Fall 2014 Sukumar Ghosh

2. Sequence A sequence is an ordered list of elements.

3. Examples of Sequence

5. Not all sequences are arithmetic or geometric sequences. An example is Fibonacci sequence

6. Examples of Sequence

7. More on Fibonacci Sequence

8. Examples of Golden Ratio

9. Sequence Formula

11. Some useful sequences

12. Summation

13. Evaluating sequences

14. Arithmetic Series Consider an arithmetic series a 1, a 2, a 3, …, a n. If the common difference (a i+1 - a 1 ) = d, then we can compute the k th term a k as follows: a 2 = a 1 + d a 3 = a 2 + d = a 1 +2 d a 4 = a 3 + d = a 1 + 3d a k = a 1 + (k-1).d

15. Evaluating sequences

16. Sum of arithmetic series

17. Solve this Calculate 1 2 + 2 2 + 3 2 + 4 2 + … + n 2 [Answer n.(n+1).(2n+1) / 6] 1 + 2 + 3 + … + n = ? [Answer: n.(n+1) / 2] why?

18. Can you evaluate this? Here is the trick. Note that Does it help?

19. Double Summation

20. Sum of geometric series Thus, So,

21. Sum of infinite geometric series So,

22. Solve the following

23. Sum of harmonic series

25. Book stacking example

27. Useful summation formulae See page 166 of Rosen Volume 7

28. Products

29. Dealing with Products

30. Factorial

32. Stirling’s formula A few steps are omitted here

33. Countable sets Cardinality measures the number of elements in a set. DEF. Two sets A and B have the same cardinality, if and only if there is a one-to-one correspondence from A to B. Can we extend this to infinite sets? DEF. A set that is either finite or has the same cardinality as the set of positive integers is called a countable set.

34. Countable sets Example. Show that the set of odd positive integers is countable. f(n) = 2n-1 (n=1 means f(n) = 1, n=2 means f(n) = 3 and so on) Thus f : Z +  {the set of of odd positive integers}. So it is a countable set. The cardinality of such an infinite countable set is denoted by (called aleph null) Larger and smaller infinities ….

35. Fun with infinite sets Hilbert’s Grand Hotel Accommodates a finite number of guests in a full hotel

36. Countable sets Theorem. The set of rational numbers is countable. Counting follows the direction of the arrows, and you cover all real numbers 1 2 34 5

37. Countable sets Theorem. The set of real numbers is not countable. (See pp 173-174 of your textbook) Proof by contradiction. Consider the set of real numbers between 0 and 1 and list them as (Here, r 1 is the first number, r 1 is the second number, and so on). But r is a new number different from the rest! So how can you assign a unique serial number to it? Create a new number r = 0.d 1.d 2.d 3.d 4 … where