User blog comment:Spitemaster/How big is this function based on the Harmonic series?/@comment-30754445-20190220190800/@comment-26693597-20190220215922

No, that's the thing. There aren't functions that aren't in this sequence which diverge slower - if they diverge any slower, they converge! Well, not unless you count functions which differ from something in this sequence by a constant. The second answer on the linked question demonstrates this: this sequence of functions converges to a function on the "boundary" between convergence and divergence.