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/
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