In my Analysis class the other day we were discussing Sequences that have bounded partial sums yet their infinite series does not converge, with the typical example of $\{ a_n \}=(-1)^n$. In discussion we started to wonder if there exists an unbounded sequence that had bounded partial sums.
I have been thinking about this for a few days and I think that $$b_n = (-1)^nln(n) $$ is such a sequence. In Mathematica I tested the first 2,000,000 sums and they are all within $\pm10 $. I would like to be able to prove that the partial sums are bounded yet I really do not know even where to start. If you can come up with any other examples or know where to start proving $$ \sum_1^n (-1)^nln(n) $$ is bounded, I would love to hear what you have to say.