Converge or Diverge: Solving ∞Ʃ (kth root of k)/k^3 using Comparison Test

[catsfanj]

Homework Statement

Ok, so I was given this problem:
∞
Ʃ (kth root of k)/k^3 = k^(1/k) / k^3
k=1

Homework Equations

None

The Attempt at a Solution

I know that I have to use the comparison test, but am unsure how to apply it. Can somebody help me?

 

[clamtrox]

You can write that into Ʃk^1/k-3. Can you find a nice real number r that has the property that r>1/k-3 for all k and Ʃk^r converges?

 

[catsfanj]

No I can't.

 

[micromass]

Can you tell us for which real numbers r, the series \sum_k k^r converges?

 

[catsfanj]

I know that the series converges, but do not know what to use for the second series. I need to prove its convergence using one of the tests.

 

[micromass]

I know that the series converges, but do not know what to use for the second series. I need to prove its convergence using one of the tests.

What series converges?? What second series??

Can you answer my question about \sum k^r?? For which r does it converge?