series vs. sequence

shahar said:
What are the differences between these two?
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In non-mathematical use, the terms are often used as synonyms, so that you might see dictionary entries like these:
  • sequence: the following of one thing after another; succession. ... a continuous or connected series: a sonnet sequence.
  • series: a group or a number of related or similar things, events, etc., arranged or occurring in temporal, spatial, or other order or succession; sequence.
That particular source doesn't even mention the mathematical usage!

But the mathematical definitions are clearly distinct, and any good dictionary should show them:
  • sequence: a continuous or connected series: such as ... a set of elements ordered so that they can be labeled with the positive integers
  • series: a number of things or events of the same class coming one after another in spatial or temporal succession; the indicated sum of a usually infinite sequence of numbers
For this reason, when you look up a mathematical term, you should look in a mathematical source:
  • sequence: A sequence is an ordered set of mathematical objects. Sequences of object are most commonly denoted using braces. For example, the symbol [MATH]\{2n\}_{n=1}^\infty[/MATH] denotes the infinite sequence of even numbers {2,4,...,2n,...}.
  • series: A series is an infinite ordered set of terms combined together by the addition operator. ...
There's a lot more that can be said, and if you search for "sequence vs. series" you should be able to find the distinction elaborated in many places. (Why didn't you?) But the basic distinction is that a sequence is a mere list, while a series is to be added.
 
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