Convergent and divergent sequences. Limit of sequence. Sandwich theorem. Rita Korsunsky

Sequences Example 1 Sequence Notation: {an} =a1, a2,a3,…,an,… A sequence is a function f whose domain is the set of positive integers 1 2 3 4 5 Sequence Notation: {an} =a1, a2,a3,…,an,… For example: {2n}= 21, 22, 23,…,2n,… Example 1 List the first four terms and the tenth term of each sequence: Sequence an a1, a2, a3, a4 a10 a. {2 + (0.1)n} 2 + (0.1)n 2.1, 2.01, 2.001, 2.0001 2.0000000001 c. {4} 4 4, 4, 4, 4 4

Recursive Definition of the Sequence A sequence is defined recursively when each term following the first term in the sequence is defined with the previous term in the sequence Example 2: Find the first four terms and the nth term of the sequence : a1 = 3 and ak+1 = 2ak for k>1

Theorem Example 3 Sequence {an} converges to 1 Let {an} be a sequence, f(n) = an, and f(x) exists for every real x>1 Example 3 Sequence {an} converges to 1

Example 4 The sequence converges to 0 Determine whether the sequence converges or diverges The sequence diverges (b) {(-1)n-1} 1,-1,1,-1,1,-1,1,… The sequence diverges Using L’Hopital’s Rule The sequence converges to 0

Theorem Example 5 Example 6 r = 1.01>1 List the first three terms of the sequence, and determine if it converges or diverges: sequence converges to 0 (b) {1.01n} r = 1.01>1 1.01, 1.0201, 1.0303 sequence diverges Example 6 Find the limit of the sequence

The Sandwich Theorem then Example 8: If {an}, {bn}, and {cn}are sequences and an<bn<cn for every n and if then Example 8: Find the limit of the sequence {cos2n/3n} 0cos2n 1 for every positive integer n, so The limit of the sequence is 0

Theorem Example 9 Let {an} be a sequence. Proof: Example 9 Suppose the nth term of a sequence is 1, -1/2, 1/3, -1/4, 1/5, -1/6, 1/7

Definitions Theorem A bounded, monotonic sequence has a limit A sequence is monotonic if successive terms are decreasing or increasing or all equal A sequence is bounded if there is a positive real number M such < M for every k Theorem A bounded, monotonic sequence has a limit