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Alternating series test states that if {$x_n$} is a decreasing sequence converging to $0$, then $\sum_{n=1}^{\infty}(-1)^{n+1}x_n$ converges. Monotonicity is important because otherwise examples such the one here can be constructed, where $\lim_{x \rightarrow \infty}x_n = 0$ but the series diverges. I wanted to know how to construct these examples. Does this have to do with rearrangement? In the link the person gives an example, but I am not sure what thinking to have when constructing these examples.

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    $\begingroup$ You want the sum of positive elements to diverge to infinity faster then the sum of negative elements or to have that sum of negative elements doesn't converge. $\endgroup$ – kingW3 Apr 1 '18 at 23:03
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No, it really has nothing to do with rearrangement. In the example you cite, the series of odd-numbered terms is divergent and the series of even-numbered terms is convergent, so the sum will clearly diverge.

Obviously, you can construct as many examples as you like by using this pattern.

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