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Infinite Series
"p" Series

 
 The p Series
 The other basic type of series is called the p Series.
 It looks like this ...
 

 "p" is just some constant.

 
 Notice that with this one we start with n = 1, not n = 0.
 Notice also that we show n in a denominator.
 The deal is, if n is in the numerator, the number will be bigger than 1.
 Adding up a bunch of these to ANY exponent greater than 0 will go to infinity
 so those are kind of no-brainers.
 Notice also that since the numerator is 1, 
 showing the exponent on the denominator or the whole fraction
 is really the same thing.
 That is ...
 

 
 Some examples of this kind of series are ...
 

 
 This type of series adds up to a number if p > 1.
 The p series looks a lot like a geometric series.
 Ones that converge to a number look really similar.
 But they are really two different things.
 

 
 The difference gets important when you calculate 
 the number to which the p Series totals.
 The value of a p Series is totaled in a completely different way.
 
 What we use is an integral.
 So for the last p series ...
 

 
 You can change the n to an X if it makes you feel better.
 

 
 We could have used an integral on the geometric example too,
 but the formula we did use is much easier.
 

   copyright 2005 Bruce Kirkpatrick

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