9.1 Sequences and Series

1. 9.1 Sequences and Series Recursive Sequence Factorial Notation Summation Notation

2. Definition of a Sequence The values of a function whose domain is positive set of integers. f(1), f(2), f(3), f(4),…f(n)....called terms written as a1, a2, a3, a4, …, an, … This is an infinite sequence since it does not end. If the sequence has a final term, it is called a finite sequence.

3. Find the first 4 terms of the sequence an = 2n + 1 n = 1 a1 = 2(1) + 1 = 3 n = 2 a2= 2(2) + 1 = 5 n = 3 a3 = 2(3) + 1 = 7 n = 4 a4 = 2(4) + 1 = 9 3, 5, 7, 9

4. Find the first 4 terms of the sequence n = 1 n = 2 n = 3 n = 4

5. Recursive Sequence A sequence where each term of the sequence is defined as a function of the preceding terms. Example: an = an -2 + an-1 : a1= 1, a2 = 1 a3 = 1 + 1 = 2 a4 = 1 + 2 = 3 a5 = 2 + 3 = 5 1,1, 2, 3, 5, …….

6. The Fibonacci Sequence is a Recursive Sequence In mathematics, the Fibonacci numbers are the numbers in the following integer sequence: By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.

7. Why are they called Fibonacci Leonardo Pisano Bigollo (c. 1170 – c. 1250) also known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages." Fibonacci is best known to the modern world for the spreading of the Hindu-Arabic numeral system in Europe,

8. Definition of Factorial Notation If n is a positive integer, n factorial is 0! = 1

9. Find the first 5 terms of the sequence Start where n = 1

10. Find the first 5 terms of the sequence Start where n = 1

11. Evaluate Expand

12. Evaluate Expand

13. Summation Notation or Sigma Notation If you add the terms of a sequence, the sequence is called a Series. Upper limit Lower limit Series Index variable

14. Find each Sum

15. Find each Sum

16. Find each Sum

17. Find each Sum

18. Properties of Sums If c is a constant

19. Properties of Sums

20. Find the Sum

21. Find the Sum

22. Find the Sum

23. The Infinite Series Infinite Series go on forever, some converge on one number, other decrease or increase forever. Does this Series ever end in one number?

24. The Infinite Series Infinite Series go on forever, some converge on one number, another decrease or increase forever. Does this Series ever end in one number?

25. Homework Page 621 – 622 # 1, 10, 17, 28, 40, 55, 58, 66, 68, 74, 77, 84, 87, 92