Download Sequences - VT Math Department

8.1 Sequences
Def: An infinite sequence of numbers is a function whose
domain is the set of positive integers
a1, a2, a3, a4, …, an, ….

a
or
a




We can denote the sequence by its nth term n
n n 1
Ex. 1, 2, 3, …, n, ….
1 1
1
1,

,
,

,....
Ex.
2 3
4

Ex. an  cosn

3n  1
a

Ex. n
n2

en
Ex. an  2 n


lim an  L where L is a
Def: A sequence {an} converges if n
finite number. If {an} does not converge, then it diverges.
Ex.
an  n


Ex. an  1
n 1

3n  1
a

Ex. n
n2

1
n
en
Ex. an  2 n

Ex.
Theorem 1 ( pg. 506): Let {an} and {bn} be sequences of real
numbers and let A and B be real numbers. The following rules
hold if
lim an  A and lim bn  B .
n
1. Sum Rule:

n
lim an  bn   A  B
n


lim an  bn  A  B
2. Difference Rule: n


lim an  bn  A  B
3. Product Rule: n
lim kan   kA
4. 
Constant Multiple Rule: n

an A
lim

5. Quotient Rule: n
b
B
n



Sandwich Theorem for Sequences: Let {an}, {bn}, and {cn} be
sequences of real numbers. If an ≤ bn ≤ cn holds for all n
beyond some index N and if
lim an  lim c n  L, then lim bn  L.
n
n
n
3  (1) n 
Ex. Determine if  n 2  converges.



n cos n 

2
Ex. Determine if 
n  1 converges.

Ex. Determine if the sequence whose nth term is an  n
converges.

Do: Do the following sequences converge or diverge?
n
1 
a


1

sin
1. n
n 
n
a 
2. n 1  n


1
n
A sequence can be defined recursively by giving:
1. The value(s) of the initial term or terms and
2. A rule, called a recursion formula, for calculating later
terms from terms that precede it.
Ex. Let a1 = 3, a2 = 5, an+2 = 2an – an+1
Write out the first five terms of the sequence.