In Mary L. Boas'(adsbygoogle = window.adsbygoogle || []).push({}); Mathematical Methods in the Physical Science, 3rd ed, on page 17 it goes over absolute convergence, and defines the test for alternating series as follows:

An alternating series converges if the absolute value of the terms decreases steadily to zero, that is, if |a_{n+1}| ≤ |a_{n}|, and lim_{n→∞}a_{n}= 0.

Now, my confusion starts with the limit. In the text book, the limit is NOT of the absolute value of a_{n}, otherwise they would have written |a_{n}|, right?

In the example given to show this definition in operation, they are using the series ∑ [(-1)^{n+1}]/n from 1 to infinity, however the limit they compute is lim_{n→∞}1/n and NOT lim_{n→∞}[(-1)^{n+1}]/n.

So did they just forget the absolute value sign in the definition (where they wrote the limit)? Or did they just feel showing the absolute value was trivial? Or do I just not understand the definition given? Does a_{n}always mean you just disregard the (-1)^{n}?

Thanks for insight.

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# B Absolute convergence Text Book question: Boas 3rd Ed

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