



Sequence
 a collection of numbers, not a sum.



1,
3, 5, 7 ... is the sequence of positive odd integers. 





A
sequence either ... 





1)
goes to either plus or minus infinity 


2) goes
to some single value 


3) flops
around between values but goes to no place in particular. 





A
sequence where the numbers keep getting bigger is called
"increasing". 


A
sequence where the numbers keep getting smaller is called
"decreasing". 


A
sequence where the numbers keep getting bigger 


or
remain the same is called "nondecreasing". 


A
sequence where the numbers keep getting smaller 


or
remain the same is called "nonincreasing". 


A
sequence which is either non increasing 


or
non decreasing is also called monotone. 





Most
if not all sequences can be written 


as
a special type of sequence notation. 


This
looks like ... 











This
looks a bit like the summation notation (S),
but remember, 


a
sequence is just a string of numbers, not a sum. 


The
numbers at the right edge of the notation 


say
that we begin with n = 1. 


So
1 is "plugged in" for n in the expression inside the
brackets 


to
give us our first sequence term. 











Now
the value of n is increased by 1 to get n = 2. 


This
"plugs in" to give us our second sequence term ... 











Motoring
along with n = 3, n = 4, n = 5 ... we get: 











Notice
that the numbers keep getting bigger. 


To
see if they always get bigger, 


treat
n as a variable and use the derivative graphing tricks 


you
learned way back when ... 











The
slope is + ½, which means the numbers always increase. 


This
means we can call this sequence increasing, 


or
the less restrictive non decreasing if we want to. 


And
since a sequence that is either non decreasing (like this one) 


or
non increasing is allowed to use the term monotone, 


we
get to use that one too! 


(lucky
us, eh?) 





Now
we know that the numbers keep getting bigger, 


but
do they go towards infinity as n gets really, really, big 


or
do they get closer and closer to some number value 


(like
when you calculate the value of e) 





We
check it with a limit ... 











So
the numbers go to infinity not some number value. 





Let's
try another one ... 





Example: 











The
derivative test (a quotient rule problem) gives us ... 











OK,
so for any positive value of n, the derivative is positive. 


So
it is increasing. 





Now
let's see where it goes. 


(Do
a limit as n approaches infinity on the original problem.) 











Multiply
through by ... 











Simplifying
... 











So
while on this one the numbers keep getting bigger, 


they
close in on 3. 


Since
they never get bigger than 3, 


3 is
called the upper bounds of the sequence. 


It is a
bit like an asymptote. 





What
is the lower bound then? 





Since
the thing is always increasing, and started at n = 1, 


the
sequence value at n = 1 must be the lower bound. 











OK,
so what kind of sequence doesn't go only up or down? 


How
about ... 











In
fact, put this 1n thing in front of most sequences and we get
trouble. 





Sometimes
the way a sequence is written 


makes
it hard to do the testing on it. 


One
strategy is to: 





1)
Find a sequence that is a bit smaller 


and
easier to analyze and see what it does. 


2)
Find a sequence that is a bit bigger 


and
easier to analyze and see what it does. 





If
the slightly smaller and slightly larger series 


both
end up at the same place, 


our
hard to analyze series ends up there two. 


The
smaller and larger series act like guides. 


This
idea is called the squeeze test. 


(Cute
name eh?) 





In
math talk we say ... 





IF 





smaller
series < our series < bigger series






AND 











THEN 











This
strategy is used a lot when we have factorials (n!) 


Like
... 











OK,
this one is simple, but it's just an example. 


We
can't do algebra on factorials too easily, 


but
algebra with exponents is easy. 


Let's
compare n! with 2 ^{n}. 






1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
n! 
1 
2 
6 
24 
120 
720 
5,040 
40,320 
362,880 
3,628,800 
2^{n} 
2 
4 
8 
16 
32 
64 
128 
256 
512 
1,024 






You
can see that after an early lead, 2
^{n} fades fast. 


That
means ... 











So
for large values of n ... 











We
can also say for large values of n ... 











Putting
this together ... 











also
... 











So
by the squeeze method ... 











copyright 2005 Bruce Kirkpatrick 
