



Back
when we talked about how to find e^{X},



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 
