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What is the correct terminology when referring to the sum of a sequence? I see people and websites use "sum of the series...", but shouldn't we say the value of the series? (Sum of a sequence and value of a series). I would go as far as say that the sum of a series is 1, because it contains 1 series.

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    $\begingroup$ Yes, in principle it should be value, but sum is well-established. $\endgroup$ – André Nicolas May 8 '14 at 2:13
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A series is the sum of a sequence. This is usually presented as an unevaluated summation of the terms of the series.

While the value of the series would be the pedantically correct parlance, the sum of the series is used to indicate evaluating that summation.

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"Sum of series" is an abuse of terminology. A series by definition is an infinite sum. A finite sum is typically called either a finite sum or partial sum. The series is the limit of the sequence of partial sums. Though redundant, people use the word infinite series to emphasize the fact that we are summing an infinite number of terms. Hence, you are completely justified in saying that we should only talk about the value of the series, than sum of the series.

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The sum of the series makes sense, since the series is defined as a sum $\sum^\infty_{k=1}a_k$. You are confusing yourself - you are mixing it up - you're thinking that a sequence (in formal terminology) is a series (in your terminology), and a series (in formal terminology) is a sequence (in your terminology).

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  • $\begingroup$ -1. This is incorrect. Series by definition is an infinite sum. $\endgroup$ – Srinivasa Ramanujan May 8 '14 at 2:24
  • $\begingroup$ @SrinivasaRamanujan I edited my answer. $\endgroup$ – user122283 May 8 '14 at 2:25
  • $\begingroup$ @SrinivasaRamanujan A series is not a sum, it is a limit. This seems to be the OPs inquiry. However, the name sum is merely informal, and doesn't refer to a sum in the usual sense. $\endgroup$ – Pedro Tamaroff May 8 '14 at 2:44

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