What is Meant by "the Sum of a Series"?

Date: 10/21/2003 at 16:18:05
From: Victor
Subject: sequences and series

What is the difference between a sequence and a series?  I think that
a series is the sum of consecutive terms, so why do many authors speak
about "the sum of a series" if the name "series" already means "sum of
consecutive terms"?

Date: 10/23/2003 at 11:00:22
From: Doctor Peterson
Subject: Re: sequences and series

Hi, Victor.

This is a good question!

We use the word "sum" in two slightly different ways.  One, which we 
might call a "formal sum" since the focus is on the form, is for an 
EXPRESSION such as

  3 + 4

which shows some number of terms being added.  The addition is only 
indicated; it has not yet been done.

The other is for a VALUE such as 7, which is the sum of 3 and 4.  The 
addition in this case has been done already, and we are concerned 
only with the result.

When we define a series as the (formal) sum of consecutive terms, we 
are thinking of the form, not the value:

  a1 + a2 + a3 + ... + an

or

  1 + 2 + 4 + ... + 32

is a sum of terms, though we have not actually evaluated it.  (As 
evidence that we have a formal sum in mind, note that we speak often 
of the "terms" of the series, which would not exist if the series 
meant the value of the sum.)  When we then talk about the sum of a 
series, we are referring to the value of this sum (63 in the second 
case); that is, we are evaluating the sum.  In the case of an infinite 
series, the sum of the series must be expressed as a limit of partial 
sums, so the evaluation process is different, but the distinction is 
the same.

The distinction is subtle!

We could reasonably talk instead of the "value" of a series; but I 
don't think I've ever heard that usage.  Perhaps we feel a need to 
reinforce the idea that the series is a sum, because it is so common 
to confuse the terms "sequence" and "series".  A series, after all, is 
essentially just a sequence with addition signs replacing the commas; 
that is, it is just a sequence that is to be summed.  A sequence in 
itself does not have a "value"; but if we have forgotten that we are 
talking about a series and not a sequence, when we talk of its "sum" 
we can't help knowing what we have in mind!

If you have any further questions, feel free to write back.

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