# Unbounded Sequence with Bounded Partial Sums

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.

Assume that your partial sums are bounded between $\pm M, M>0$
Then since your sequence is unbounded(WLOG from above) there would be an $a_{n_0} >3M$
Then if you take $$\sum _{i=1}^{n_0}a_i = \sum _{i=1}^{n_0-1}a_i+a_{n_0} >-M +3M > 2M$$ a contradiction