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!
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