BC 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 , if we can
n 
force an to be as close to L as we wish by choosing n large enough. If {an } does not converge,
we say the sequence {an } diverges.
For each of the following sequences i. write out the first five terms of the sequence, and ii.
determine with explanation whether the sequence converges or diverges . 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 14
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 5 7
, , , ,
3 9 27 81
4.
1 5 23 119
, , ,
,
2 6 24 120
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
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!

11.
xn
Shift the index of 
so that the index starts at one instead of zero.
n  0 2n  1
IMSA BC3
Sequences and Series 1 p.2
Fall 14
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

Do the following problems on pages 467-69: 20,25,26,38,43
IMSA BC3
Sequences and Series 1 p.3
Fall 14
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.
Changing the sense of the inequalities above we can similarly define decreasing and strictly
decreasing 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 14
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) If {an  bn } is convergent, then {an } and {bn } are convergent.
IMSA BC3
Sequences and Series 1 p.5
Fall 14
Some more challenging fun problems.
k2
(1) Evaluate lim an , where an   3 .
n 
k 1 n
n
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
Here are some problems from the text that you will find amusing.
Pages 467-69: 19,22,23,25,26,27,31,38,43,51,57
IMSA BC3
Sequences and Series 1 p.6
Fall 14