I'd like to suggest a start but not sure it's sound. Perhaps others can comment.

First, clearly define the integral:

so need to consider:

Now express the last expression as the same type integral:

Combining both sums, I obtain:

Now, at this point, how about using the Integral Test? I'm however not sure that's valid and I'd need to work on that part. If it were valid, I could then write:

Switching the order:

I get:

and that last term diverges. So if the steps I've taken so far are valid, and I could apply the integral test in this fashion, then I would conclude the original sum diverges.

. . . Gee whiz, you guys are way better at this than me but I'm big at tries.