Seems to me that you can easily show that this diverges from the original definition. I presume that your numerical calculations (I'm not sure what you mean by "analytically". If you have show analytically that this diverges, you are done.) diverges to negative infinity.
That should lead you to look at (1/2)ln(1/n)< -M for M some large integer. Then ln(1/n)< -2M so 1/n< e-2M[/sum] and n> eM. Working the other way, for any integer M, if n< eM then (1/2)ln(1/n)< -M and so the sequence diverges.