1. The problem statement, all variables and given/known data Hi everybody! I'm stuck on a problem in which I don't manage to prove absolute convergence (or not) of the series ∑ (-1)j/(√j + √(j+1)). 2. Relevant equations Leibniz criterion, minorant criterion, limit comparison test, maybe others 3. The attempt at a solution So first I noticed that the series alternates, so I ran the Leibniz test in order to find out if the series converges/diverges: |an+1| - |an| = 1/(√(j+1) + √(j+2)) - 1/(√j + √(j+1)) = (√j - √(j+2))/[(√(j+1) + √(j+2))⋅(√j + √(j+1))] < 0 ⇒ the sequence is monotone decreasing. lim j→∞ (1/(√j + √(j+1))) = 0 ⇒ the series converges. Now that was rather easy, but I know that in order to prove that the series absolutely converges or not I must find out if the sequence |an| converges or diverges. And that's my problem. I imagine it being divergent, but I cannot seem to find a known divergent series smaller than 1/(√j + √(j+1)) (which btw can be rewritten as (√(j+1) - √j)). Any hint? Maybe I'm just using the wrong criterion. I just tried the limit comparison test, which didn't work, but that might be because I never used it (somehow we don't have this criterion in our class though it seems very important!!). Any advices about that test? How do you choose your "other series" to run the test with? Thank you very much in advance for your answers. Julien.