Root test got a 1, I wasn't able to do the ratio test (algebraic ugliness never cleared up), we can't integrate it (right?), I don't know what to compare it to, so the freshman calculus stuff isn't working.
I thought about factoring out a piece like and applying Dirichlet's test (but the factored out part doesn't go to zero) or Abel's test (but the remaining ^(n-1) part may or may not converge.)
I thought about (abstractly) converting this to a power series, calling it equal to the original sum, and taking derivatives to see if I could find something easier to identify a radius of convergence, but with the power series I think I get into conditional convergence and the effect of rearranging terms..
[EDIT: You may skip some inconclusive dialog and get to my latest attempt to prove divergence here at post #11.]