Interesting problem from Strang's Calculus I'm having thoughts on.

Every second a computer adds a million terms of the divergent series with general term 1/(n ln n) (defined for n > 1). By comparison with the associated integral, estimate the partial sum after a million years.

I know that the number of terms counted after a million years is about 3.2 x 10^{19}. I'm confused onto how to proceed. Are there any suggestions? Thanks for any help with such a great problem!