Sequence. A list of numbers written in a definite order.

Transcription

1 Sequence A list of numbers written in a definite order.

2 Terms of a Sequence a n = 2 n 2 1, 2 2, 2 3, 2 4, 2 n, 2, 4, 8, 16, 2 n We are going to be mainly concerned with infinite sequences. This means we let n.

3 Definition of a Sequence a n = 2 n 2 1, 2 2, 2 3, 2 4, 2 n, 2, 4, 8, 16, 2 n A sequence is simply a list of numbers with an identifiable first member, second member, third member etc. Mathematically, a sequence is defined as a function whose domain is a set of positive integers.

4 Write the first 4 terms for the following sequence.

5 Find a formula for the general term a n of the sequence. * 1 n+1 will also work.

6 Explicit vs Recursively Defined Sequence All of the previous sequence have been defined explicitly because a n is defined in terms of n. a n = 2 n Another way to define a sequence is recursively. In this definition, you use a previous term to define a n. a n+1 = a n 2

7 Recursively Defined Sequence A sequence whose terms are defined using previous terms. d n+1 = d n 5 d 1 = 25 d 1 = 25 d 2 = 25 5 = 20 d 3 = 20 5 = 15 d 4 = 15 5 = 10 *Note the first term of a recursively defined sequence must be given.

8 What are the first four terms of the recursive sequence? a n+1 = a n 2 5 a 1 = 3 3, 4, 11, 116

9 Fibonnaci Sequence The Fibonnaci Sequence is probably the most famous recursive sequence. The sequence was used when studying a population of rabbits by a man named Fibonnaci who lived from ( ) in Italy. a n+2 = a n + a n+1 a 1 = 1 a 2 = 1 What are the first 12 terms of this sequence? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

10 What is the difference between these two? a n = 2 n y = 2 x Domain is only the set of positive integers (1, 2, 3, 4.) Domain is all numbers (, )

11 Graphing a Sequence with a Calculator Press Mode, then change function to sequence. (Be sure to press enter to save) Now Press y= and you will see u instead of y. Input n+1 (use alpha>6 for n) and then hit graph. *Note u is a function of n and not x.

12 Arithmetic vs Geometric Sequences Arithmetic Sequences have a common difference. 1, 4, 7, 10, 13, 16, Common difference is 3 Geometric Sequences have a common ratio. 1, -2, 4, -8, 16,.. Common Ratio is -2

13 Arithmetic Sequence An arithmetic sequence is one with a common difference between terms. 1, 4, 7, 10, 13, 16, Common difference is 3 Arithmetic Sequences can generally be defined both explicitly and recursively. Explicit a n = 1 + 3(n 1) = 3n 2 First Term Common Difference Recursive a n+1 = a n + 3 a 1 = 1

14 Geometric Sequence A Geometric Sequence is a sequence with a common ratio between consecutive terms. 1, -2, 4, -8, 16,.. 10, 5, 5 2, 5 4, 5 8,. Common Ratio is -2 Common Ratio is 1 2 If r is the common ration and a is the first term, a Geometric Sequence can be written as: a n = ar n or a n = ar n 1 a n = 1 ( 2) n 1 a n = n 1

15 Factorials 5! n! n + 3! 2n 1! n 1! n + 2!

16 Homework Section 9.1 (1, 5, 13, 15-26, 29-31, 33, 35) *For draw a rough sketch and label *For problems 25, 26, 29, 30, Tell me: a. Is it a geometric or arithmetic sequence b. Find a n term. If arithmetic, find both an explicit and recursive formula*

17 Limit of a Sequence What happens to the sequence of numbers as n?

18 Limit of a Sequence If the limit of a sequence exists then the sequence converges. If the limit of a sequence does not exist (terms go to ± or alternating sequence) then the sequence diverges.

19 Do these sequences converge or diverge? Converge Diverge Converge Converge Diverges to Infinity Diverges to Negative Infinity

20 If a function f(x) and a sequence a n are equal for all positive integers lim x x 1 2x *Note that these functions are the same except for in their domains.

21 Do the following Sequences Converge or Diverge? If they converge, what do they converge to? a n = 1 n 1 lim n n Converge to 0 a n = n 1 5n lim n n 1 5n Converge to 1 5 a n = 2 n 2, 4, 8, 16, 32, 64,.. Diverges to infinity n a n = , 1 4, 1 8, 1 16, 1 32, 1 64,.. Converges to 0 a n = 1 n Diverges 1, 1, 1, 1, 1, 1,.. a n = ln n n ln n lim n n L Hopital s 1 lim n n Converges to 0

22 The Squeeze or Sandwich Theorem L

23 Show that the sequence cos n n converges and find its limit. 1 cos n 1 1 n cos n n 1 n 1 lim 1 lim n n = 0 = 0 n n lim n cos n n = 0 The limit as n of cos n n converges to 0.

24 a n vs n! Which of the two will grow faster? 2 n or n! n 4 3 n = = n n 7 = It can be seen that in both cases, n! will eventually grow faster. This will hold for all a n and as n, n! Will always grow faster than a n.

25 n 5, 5 n, n! For what values of n will each be the largest? Graph each and use the table on your calculator to save time. n 5 will be larger only when 2 n < 5 5 n will be larger only when 5 < n 11 n! will be larger when 12 n

26 Monotonic Sequence A sequence a n is monotonic if its terms are nondecreasing. Or if its terms are nonincreasing. 1, 2, 3, 3, 4, 5, 6, 7, 8 1 2, 1 4, 1 8, 1 16, 1 32, 1 64,.. 1, 2, 3, 3, 4

27 A Special Recursive Sequence a n+1 = 1 a n + a n a 1 = 1 2

28 A sequence to estimate Square root of 2 as a sequence

29 Homework Section omit (39, 41) odd omit (49, 51) odd, 59, 80, 85, 93