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.