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The alternating series test says that an alternating series whose terms are monotonically decreasing in absolute value and which approach zero in the limit converges.

Of course this is not an if-and-only-if statement -- most convergent alternating series do not have monotonically decreasing terms -- but if $a_n$ is an alternating series satisfying the hypothesis of the test, is there any term shorter than "alternating series with terms monotonically decreasing in absolute value which approach zero in the limit?"

For the past decade or so I thought that there was such a term and that the term was "telescoping series." This would seem to make sense, because the partial sums of such a series telescope in on themselves. Apparently, however, "telescoping series" has something to do with the formulas for the partial sums, and not for the partial sums themselves.

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Typically we refer to an alternating series that converges as "conditionally-convergent". Of course if the series converges absolutely then it is also absolutely convergent. Naturally, there are some of the former that aren't the latter.

Telescoping refers to when the partial sum terms all cancel except for perhaps a couple of the "end-terms". For example $\sum(\frac{1}{n}-\frac{1}{n+1})$ is telescoping.

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  • $\begingroup$ I'm asking specifically about the ones where $|a_n|$ decreases monotonically, i.e. the ones satisfying the conditions of the alternating series test. $\endgroup$ – Daniel McLaury Dec 4 '15 at 20:46

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