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Sum of An Infinite Series


Date: 07/08/98 at 13:39:37
From: Melissa May
Subject: The sum of an infinite series- how is it possible?

Hello, Dr. Math. 

Isn't adding up all the terms of an infinite series not possible?   
How can one take infinity, something so endless and flowing with 
continuity, and box it up in something so simple as a sum?  And yet we 
are presented with a formula for this very task, adding up infinity. 
Please explain this concept to me, if you can.


Date: 07/08/98 at 15:44:15
From: Doctor Rob
Subject: Re: The sum of an infinite series- how is it possible?

You are right that the way addition is defined, we can only add up a 
finite number of numbers. Any attempt to add up an infinite number of 
numbers must rely on a new definition or concept.

As an example, let's take the following infinite series:

   S = 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/2^n + ...

Consider the sequence of finite sums gotten by taking the first n 
terms:

   P(n) = 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/2^n

P(1) = 1/2, P(2) = 3/4, P(3) = 7/8, P(4) = 15/16, and so on. It can be
easily proved by mathematical induction that P(n) = 1 - 1/2^n for all
n > 0.  Now as n grows without bound, we define the sum S to be the 
value (if any) that P(n) approaches more and more closely. In the 
study of calculus, you will learn that this is called a limit. Series 
for which this limit exists are called "convergent" series. S is the 
limit as n grows without bound of P(n), written:

   S =    lim      P(n)
       n->infinity

The formal definition of a limit is not important here. You will learn
it in calculus. In this case, P(n) approaches 1 as n grows without 
bound. If you tell me how close to 1 you want to get, I can tell you 
the value of n you will have to take so that for that n and all larger 
ones, P(n) is that close or closer to 1. Thus we write S = 1.

Now it turns out that defining S that way makes a lot of sense. Such a
definition allows one to make sense of some infinite sums, and 
furthermore, convergent series obey the usual laws of arithmetic.  
Convergent series include as a special case all finite sums, by 
appending to the end an infinite string of the form:
 
   + 0 + 0 + 0 ...

and the value obtained by the above process on the infinite sum agrees 
with the value obtained by the usual methods of adding finite sums.

Some infinite series have no sums. They are called "divergent" series. 
There are two kinds of divergent series. First there are the kind where 
P(n) grows beyond all bounds, as in 1 + 1 + 1 + 1 + ..., for which 
P(n) = n. Then there are the kind where P(n) oscillates, such as 
-1 + 2 - 2 + 2 - 2 + ..., for which P(n) = (-1)^n. In the first case, 
we can write S = infinity, but have to be very careful with that kind 
if we try to manipulate them using the laws of arithmetic. After all, 
infinity is not a number in the same way that 4, -17, 3/2, sqrt(2), or 
pi is! In the second case, we cannot assign any sensible value to S at 
all.

In this particular example, we could compute the value of S in a 
different way. Multiply the sum by 2, to get:

   2*S = 2/2 + 2/4 + 2/8 + 2/16 + ... + 2/2^n + ...
       = 1 + 1/2 + 1/4 + 1/8 + ... + 1/2^(n-1) + ...

Now subtract the original series from this one. Note that almost all 
of the terms cancel, such as 1/8 - 1/8, and 1/1024 - 1/1024 (is it okay 
to cancel infinitely many of these pairs?), and you are left with:

   S = 2*S - S = 1 + 0 + 0 + 0 + ... = 1

I hope this helps clarify what we mean by the sum of an infinite 
series.

- Doctor Rob, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
    
Associated Topics:
High School Logic
High School Sequences, Series

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