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The definition of the Leibniz series convergence criteria states:

The alternating series test then says if $\{a_n\}$ decreases monotonically and goes to $0$ in the limit then the alternating series converges.

Leibniz Criterion on Wikipedia
Question: What happens if I encounter a monotonic increasing sequence which also approaches to $0$ ?
Does the test work for this increasing sequence? At least it doesn't diverge, we had an example for $a_n=(-1)^n\left(-\frac{1}{n}\right)$
According to Wolfram Alpha, this converges, but why and can I conclude that it is equal whether the sequence is monotonically decreasing/increasing?

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Your sequence $\frac1n$ is actually monotonically decreasing towards $0$.

You would need a negative sequence to increase towards $0$, but then it is still true that the absolute values should form a monotonically decreasing sequence converging to $0$.

If $\sum(-1)^na_n$ converges, then also $\sum (-1)^n(-a_n)$ and vice versa.

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  • $\begingroup$ I just see that I typed the wrong example, it must be $-\frac{1}{n}$. So in fact, it doesn't matter, as long as the absolute values form a monotonically decreasing sequence? $\endgroup$ – Christoph S Mar 28 '15 at 9:20

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