Alternating Series Test: Understanding the First Condition

[LumenPlacidum]

The alternating series test contains two conditions for convergence.

The first condition is that the nth term (extracting the power of -1) is always positive and monotonically decreasing.

The second is that the limit of that nth term goes to 0 as n goes to infinity.


I've seen a proof for it, and I've even proved it myself (although some years ago). I don't really understand why the first condition is so restrictive. It seems like the always-positive part is all you need combined with the limit going to 0.

Can someone show me an example of an alternating series for which the terms (again, without the power of -1) are always positive and that have a limit of 0, but which is NOT monotonically decreasing and diverges because of it?

 

[jostpuur]

<br /> 1,\; -1,\; \frac{1}{2},\; -\frac{1}{2^2},\; \frac{1}{3},\; -\frac{1}{3^2},\; \ldots<br />

The sum of the positive parts tends to push the series towards +\infty, and the sum of the negative parts tends to push the series towards -\pi^2/6, so the negative parts cannot cancel the positive parts sufficiently for the series to converge.

 

[LumenPlacidum]

Great, thanks so much!