How can I apply the Leibniz test if an alternating series doesn't decrease?

[Sophia_M]

I’ve been working through some problems on the convergence of infinite series, and I keep getting stuck when a series has terms that alternate in sign but don’t strictly decrease in absolute value from the very first term. It makes applying the Leibniz test for alternating series feel a bit shaky to me.

[Jeffrey83]

Yeah I ran into this last semester. If a_n is positive and goes to 0 but isn’t decreasing from the first term, it’s enough to know it is eventually decreasing. Once you reach some N so that a_{n+1} <= a_n for all n >= N and a_n -> 0, the alternating series test tells you the tail converges, and the finite head doesn’t affect convergence.

[Ronald61]

I had a series where the terms bounced around at the start. I checked the tail from N=10 where it settled, and the partial sums of (-1)^n a_n from there looked stable; the full sum seemed to converge, but I wasn’t sure about the early terms.

[GregoryZL]

Do you sometimes worry the problem is actually about whether the convergence is conditional rather than absolute, or about applying a rearrangement or a Dirichlet style test?

[JosephMG]

I tried a grouping trick once, pairing terms: write (-1)^n a_n as (a_{2k}-a_{2k+1}) and see if the pairwise sum converges. It helped in some cases, but not all, and I ended up stuck again.