In my language, two sequences are adjacent if the first is increasing, the second is decreasing, and their difference converges to 0. I couldn't find anything online about it; what's the name for that concept in English?
Also, what are the properties of adjecent sequences, what importance does it have to label them as such?
I have a problem where $u_n = \sum^{n}_{k=1} \frac{1}{k^2}$, and $v_n = u_n + 1/n$
I want to prove that these series are adjacent, but I don't see how either of them could be decreasing; with every greater n, we're adding a positive number, no? Where am I going wrong, conceptually and mathematically? Thanks.