Difference between a Sequence and a Series

Date: 01/05/2003 at 14:51:50
From: Jessica
Subject: What is the difference between a sequence and a series?

If 45, 25, 5... is an example of a sequence, could you give me an 
example of a series?

I think both are the same; I can't tell the difference.

Date: 01/05/2003 at 16:56:12
From: Doctor Pete
Subject: Re: What is the difference between a sequence and a series?

Hi Jessica,

A sequence is a list of numbers. The order in which the numbers are 
listed is important, so for instance

     1, 2, 3, 4, 5, ...

is one sequence, and

     2, 1, 4, 3, 6, 5, ...

is an entirely different sequence.

A series is a sum of numbers. For example,

     1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

is an example of a series. A series is composed of a sequence of 
terms that are added up. The order in which the terms appear is 
sometimes important; there are certain kinds of series in which any 
rearrangement of the terms will not affect its sum (these are called 
"absolutely convergent"). There are others in which different 
arrangments of the terms will cause the sum of the series to change.

Sequences and series are often confused because they are closely 
related to each other. But when we say "sequence," we are not 
concerned with the sum of the values of the terms, whereas in a 
series, we are interested in such a sum. For example:

     1, 1/2, 1/3, 1/4, 1/5, 1/6, ...

is a sequence, and it is pretty obvious that the n(th) term will be

     1/n.

One question we may ask is, "what is the eventual behavior of the 
sequence?"  In other words, what do the values of the sequence tend 
toward as we compute more and more terms? Clearly, we see that the 
1000(th) term will be 1/1000, and the millionth term will be 
1/1000000. Eventually these terms tend towards zero, as n grows larger 
and larger. However, it is important to notice that no term of the 
sequence is ever *exactly* zero, nor is any term ever less than zero.  
We say that the "limit" of the sequence is equal to zero.

For every sequence, there is an associated series, and vice versa.  
So the associated series to this sequence is the series

     1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ...

which is simply the sum of all the terms of the sequence we previously 
described. We are now interested in whether this series has a sum; in 
other words, if we add up all the terms, is the result a number, or is 
it too large (i.e., is the sum infinite)?  The answer, which is 
perhaps beyond the scope of this discussion, is that this sum is 
"divergent."  That means it does not have a finite value, or more 
precisely, I can say the following: If you give me any number, as 
large as you please, I can tell you how many terms of this series, if 
added up, will exceed the number you gave me. For instance, if you 
said, "100,"  I would say something like 3^100 (three raised to the 
100th power). If you added that many terms together, the sum would be 
larger than 100.

A series, in turn, can be associated with the sequence of its terms; 
for instance,

     1 + 1/4 + 1/16 + 1/64 + ...

can be associated with

     1, 1/4, 1/16, 1/64, ...

which are simply the terms without the addition signs. But it may also 
be associated with another sequence, called the "sequence of partial 
sums":

     1, 5/4, 21/16, 85/16, ...

in which the n(th) term of the sequence is the sum of the first n 
terms of the series. So we have

       1 = 1,
     5/4 = 1 + 1/4,
   21/16 = 1 + 1/4 + 1/16,
   85/16 = 1 + 1/4 + 1/16 + 1/64, ...

and so on. Then it is not too hard to see that the limit of this 
sequence is equivalent to the sum of the series.

There are many interesting properties of sequences and series, some 
of which I have introduced here, but the basic difference between the 
two is that a sequence is simply a list of numbers, whereas a series 
is a sum of numbers.

- Doctor Pete, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/05/2003 at 17:52:56
From: Jessica
Subject: Thank you (What is the difference between a sequence and a 
series?)

Thank you for the answer. It helped a lot!