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A divergent alternating series whose terms go to zero

From the alternating series test, you know that if tex2html_wrap_inline375 and if tex2html_wrap_inline377 decreases monotonically to zero, then tex2html_wrap_inline379 converges. However, it is not enough to have tex2html_wrap_inline377 having a limit of zero, you also need tex2html_wrap_inline377 decreasing, as the following example shows.

Take your favorite convergent series with positive terms, say tex2html_wrap_inline385 , and take your favorite divergent positive term series whose terms go to zero, say tex2html_wrap_inline387 . Now ``shuffle'' these together to form the following series:


This alternates, the terms go to zero, but the terms are not decreasing monotonely to zero. This series also diverges. The divergent harmonic series overpowers tex2html_wrap_inline385 to force the sum off to tex2html_wrap_inline393 . (This can be made rigorous by looking at partial sums).

Tom Vogel
Mon May 5 12:53:33 CDT 1997