Re: Does the series 1 - 1/2 - 1/3 + 1/4 + 1/5 - 1/6 - 1/7 + 1/8 + 1/9 + ... converges?
There is a nice exercise in Palka's "An Introduction to Complex
Function Theory" which answers this question :
Given a sequence <z_n> with bounded partial sums, and a sequence <r_n>
of decreasing positive numbers with limit 0, then the sum
$\sum_{n=0}^\infty z_n r_n$ converges.
The prrof is a nice application of summation by parts, and extends the
classical Alternating Series Test.
In this case, take <z_n> = <1,-1,-1,1,1,-1,-1,1,1,...>, whose partial
sums are clearly bounded.
.
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