I know that 'series' like these converge (and I apologize for formatting) sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + ...)))) (with 1 going on forever), for any constant they converge. sqrt(n + sqrt(n + sqrt(n + sqrt(n + ...)))). This can be solved by setting x = the value = sqrt(n + x) and solving for x.
But I am wondering about this series. sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ...)))).
It has been a long time since I have looked at this sort of thing, and I can upper bound it by sqrt(1)+sqrt(sqrt(2))+sqrt(sqrt(sqrt(3))) + ... which diverges, so does not help, and I've thought about ratio tests and things like that, but that requires an actual series , which is hard to describe. We could set something up like a limit sequence, a_1 = sqrt(1), a_2 = sqrt(1+sqrt(2)), a_3=sqrt(1+sqrt(2+sqrt(3)), etc, and look at a_n as n approaches infinity. But I'm out of ideas on what to do here. Checking on a computer, it does appear to converge, but I'm wary of something like the harmonic series, which grows very slowly. Can anyone give me any pointers on this?
Thanks.
