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See the Horizon
Infinite Sequence

 

 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 "non-decreasing".
 A sequence where the numbers keep getting smaller
 or remain the same is called "non-increasing".
 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 + , 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
2n 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

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