BC 2-3
Patterns
Name: ____________________
Definition: An infinite sequence is a real valued function with domain
D  {n  n  M , where M is some fixed finite integer} .
Remarks:




We typically look at a sequence as a list: a1 , a2 , a3 ,
Though a sequence is a function, we normally use an instead of the conventional a(n) .
We like to use i,j,k,m,n to denote integer variables and continue to use x,y,z,s,t to
represent variables that take on some continuous set of real numbers.
We use an to represent a single term in the sequence (namely the nth term); we use the
notation {an }n 1 to represent a sequence with domain
Definition: We say the sequence {an } converges to L 
.
, denoted lim  an   L , iff for each
n 
  0 there exists a real number M such that an  L   whenever n  M . If {an } does not
converge, we say it diverges.
Essentially we are saying that we can force an to be as close to L as we wish by demanding we
are past a certain point in the sequence.
Determine (with or without proof) whether the following sequences converge or diverge. If it
converges, state the limit.
1. an 
n 1
n
2. bn  tan 1 (n)
2
3. cn   
3
n
1
4. an  n sin  
n
IMSA BC3
Sequences and Series 1 p.1
Fall 12
For each of the following patterns, find a formula for the general term.
1.
3, 6, 12, 24, 48, ….
2.
2, -5, 8, -11, 14, ….
3.
1/3, 3/9, 5/27, 7/81, ….
4.
1/2, 5/6, 23/24, 119/120, ….
5.
2, 24, 246, 2468, ….
6.
1, 13, 135, 1357, ….
Convert each of the following to summation (sigma) notation.
7.
1 – 3 + 5 – 7 + … + 201 – 203
9.
1
11.
Shift the index of
x x 2 x3
  
2 3
4
8.
1
1
1



1 2 2  3 3  4
10.
x
x3 x5 x 7
  
3! 5! 7!

IMSA BC3
xn
so that the index starts at one instead of zero.

n  0 2n  1
Sequences and Series 1 p.2
Fall 12
Note: Let f be a function defined on [a, ) for some real number a. If lim f ( x)  L , then
x 
lim f (n)  L .
n 
Perhaps, this is most useful in that it allows us to use many of the theorems developed for the
limit of functions in determining the limits of sequences. For example (but not limited to)
ˆ
rules, the squeeze theorem, sum and product rules, etc..
l'Hopital's
Determine each limit or show that the limit does not exist.
1. lim  n  e n 
n 
2. lim  n 2  tan 1 (n)   / 2  
n 
k
3. lim  2k  3k  
k  

n

4. lim   e x dx 
n 
0

IMSA BC3
Sequences and Series 1 p.3
Fall 12
Definition: Let {an } be a sequence such that an  an 1 for all values of n in the domain. Then the
sequence {an } is said to be increasing. If an  an 1 for all values of n in the domain we say that
the sequence is strictly increasing. If there is an integer M such that an  an 1 for all n in the
domain with n > M, then the sequence is eventually increasing.
Changing the sense of the inequalities above we can similarly define decreasing, strictly
decreasing, and eventually increasing sequences. A sequence that is increasing or decreasing is
said to be monotone.
Definition: Let {an } be a sequence and suppose that there is a real number M such that M  an for
all n in the domain, then we say {an } is bounded above, Similarly, there is a real number M such
that M  an for all n in the domain, then we say {an } is bounded below. If a sequence is bounded
above and below, we simply say that the sequence is bounded.
(1) Show that the sequence defined by a1  0; ak  6  ak 1 is bounded above.
One might reasonably ask, “So?” To that Bernard Bolzano might reply with the following theorem,
Theorem: Every increasing sequence that is bounded above is convergent.
The Bolzano-Weierstrass Theorem (aka The Fundamental Theorem of Bounded Sequences)
actually states that every bounded sequence has a convergent sub-sequence.
(2) Now you can prove that the sequence defined by a1  0; ak  6  ak 1 is convergent and
find its limit.
IMSA BC3
Sequences and Series 1 p.4
Fall 12
Label each statement below as true or false, If false, give a counter-example.
(1) Every convergent sequence is monotone.
(2) Every convergent sequence is bounded.
(3) If {an } and {bn } are convergent, then {an  bn } is convergent.
(4) If {an } and {bn } are convergent, then {an  bn } is convergent.
(5) Every unbounded sequence is monotone.
(6) Let f : (, ) 
convergent.
be a real-valued function. If {xn } is convergent, then { f ( xn )} is
(7) If {an  bn } is convergent, then {an } and {bn } are convergent.
Some more challenging fun problems.
k2
.
3
k 1 n
n
(1) Evaluate lim an , where an  
n 
IMSA BC3
Sequences and Series 1 p.5
Fall 12
n
i.
1
.
k 1 k  n
Find a1 , a2 , and a3 .
ii.
Show that an  1 for all n
iii.
Show that an  is increasing.
(2) Define {an } by an  
.
2
iv.
Show that lim an  ln 2 . Hint:
n 
1
 x dx  ln 2
1
(3) Let an 
n
n!
.
n
i.
Show that ln an 
ii.
Evaluate lim an
IMSA BC3
1 n
1 n
ln
k


 ln n .
n k 1
n k 1
n 
Sequences and Series 1 p.6
Fall 12