
Sequences 
Stated formally, an infinite sequence, or more simply a sequence, is an unending succession of numbers, called terms. EXAMPLES: FAMOUS SEQUENCES: 1. Prime numbers: 2. Fibonacci numbers: 3. Catalan numbers: DEFINITION: A sequence is a function whose domain is a set of integers. Specifically, we will regard the expression to be an alternative notation for the function EXAMPLE: means EXAMPLE: means EXAMPLE: means Finding Terms of Sequences
EXAMPLE: Write the first five terms of the sequence EXAMPLE: Write the first five terms of the sequence EXAMPLE: Write the first five terms of the sequence EXAMPLE: Write the first five terms of the sequence EXAMPLE: Write the first five terms of the sequence EXAMPLE: Write the first five terms of the sequences and Finding Explicit Formulas for
Sequences
EXAMPLE: Find an explicit formula for the sequence of the form with the initial terms EXAMPLE: Find an explicit formula for the sequence of the form with the initial terms EXAMPLE: Find an explicit formula for the sequence of the form with the initial terms EXAMPLE: Find an explicit formula for the sequence of the form with the initial terms EXAMPLE: Find an explicit formula for the sequence of the form with the initial terms EXAMPLE: Find an explicit formula for the sequence of the form with the initial terms EXAMPLE: Find an explicit formula for the sequence of the form with the initial terms EXAMPLE: Find an explicit formula for the sequence of the form with the initial terms EXAMPLE: Find explicit formulas for the sequences of the form with the initial terms (a) and (b) (c) (d) (e) (f) (g) Limit
DEFINITION: A sequence has the limit and we write if we can make the terms as close to as we like by taking sufficiently large. If exists, we say the sequence converges (or is convergent). Otherwise, we say that the sequence diverges (or is divergent). DEFINITION: A sequence has the limit and we write if for every there is a corresponding integer such that DEFINITION: means that for every positive number there is an integer such that THEOREM: Suppose that the sequences and converge to and respectively. Then THEOREM (The Squeeze Theorem for Sequences): If for and then THEOREM: If then REMARK 1: Note that if and then is not necessarily equal to For example, if then but does not exist. REMARK 2: The converse of the Theorem is also true, that is, if then EXAMPLE: Let then since EXAMPLE: Determine whether the sequence converges or diverges. If it converges, find the limit. EXAMPLE: Determine whether the sequence converges or diverges. If it converges, find the limit. EXAMPLE: Determine whether the sequence converges or diverges. If it converges, find the limit. EXAMPLE: Determine whether the sequence converges or diverges. If it converges, find the limit. EXAMPLE: Determine whether the sequence converges or diverges. If it converges, find the limit. EXAMPLE: Determine whether the sequence converges or diverges. If it converges, find the limit. EXAMPLE: Determine whether the sequence converges or diverges. If it converges, find the limit. EXAMPLE: Determine whether the sequence converges or diverges. If it converges, find the limit. EXAMPLE: Determine whether the sequence converges or diverges. If it converges, find the limit. EXAMPLE: Determine whether the sequence converges or diverges. If it converges, find the limit. EXAMPLE: Determine whether the sequence converges or diverges. If it converges, find the limit. EXAMPLE: Determine whether the sequence converges or diverges. If it converges, find the limit. Increasing and Decreasing Sequences DEFINITION: A sequence is called increasing if for all that is, It is called decreasing if for all that is, A sequence is monotonic if it is either increasing or decreasing. EXAMPLE: The sequence is decreasing (and monotonic), since EXAMPLE: Show that the sequence is increasing. DEFINITION: A sequence is bounded above if there is a number such that It is bounded below if there is a number such that If it is bounded above and below, then is a bounded sequence. MONOTONIC SEQUENCE THEOREM: Every bounded, monotonic sequence is convergent. 