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Infinite Series
Geometric type

 

 Back when we talked about how to find eX,

 we talked about infinite series.
 An infinite series is an infinite number of terms added together.
 If, after a few terms, each term that is added to the total
 is much smaller than the last term that was added
 the series "homes in" on some number.
 If the terms that are added don't get smaller fast enough
 the total keeps getting larger and larger as we add terms.
 The sum will go towards infinity.
 
 The terms are usually numbered from zero or one to infinity.
 The term number (n) usually shows up somewhere in the term.
 As the terms go along, n gets bigger and bigger.
 Where the n lives in the term, 
 determines if these terms are big numbers or not.
 
 If the n is a constant or part of a numerator 
 or an exponent of a constant bigger than one,
 as n gets big the terms are PROBABLY big numbers.
 And the series PROBABLY approaches infinity.
 
 If the n is a denominator or an exponent on a constant smaller than one
 as n gets big the term are PROBABLY tiny numbers
 and the series PROBABLY approaches something close to the number
 that the first few terms add up to.
 
 This chapter gives you some tricks to tell which is happening.
 
 An infinite sum is a lot like any other sum.
 We can use the summation notation (S) to represent it.
 The one difference is that the number on the top of the S,
 the upper limit, is infinity and not some number.
 
 The stuff being summed can have all kinds of weird stuff going on.
 But for the most part, it falls into one of two basic types:
 Geometric Series or "p" Series
 In a geometric series, the n is part of an exponent.
 In a "p" series, the n is part of something other than an exponent.
 
 In either one of these there can be other constants all over the term.
 They are just there to try to hide what's going on.
 Like in other calculus stuff, 
 most of the time these constants can just be factored away.
 
 Geometric Series
 The general form of this type is ...
 

 
a is just some number, that is, a constant.
 
r is a constant too, but raised to the n power
 
 a geometric series looks like ...
 

 
 So how do you tell if a series adds up to some number as n gets really big?
 
 With a geometric series, it's really easy.
 If |r|<1, it adds up to a number, otherwise it goes to infinity.
 Now the tough part.
 When |r|<1, what number does it add up to?
 
 The answer is ...
 

 
 The second geometric series example we used before was ...
 

 
 This was the only one of those three examples where |r|<1 (it's 1/8).
 So this is the only one of those examples that adds up to a number.
 The number is ...
 

 

   copyright 2005 Bruce Kirkpatrick

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