In the best tradition of noding your homework, here we go:

If basic comparison tests on infinite series do not let you conclude on convergence, you can try Riemann condition.

To be applicable, you need a sequence **u**_{n} positive (**u**_{n}≥ 0 for all **n**).

If you can find α > 1 so that:

**(n**^{α}u_{n})_{n ∈ N} has an upper bound

then **Σu**_{n} converges.

If you can find α ≤ 1 so that:

**(n**^{α}u_{n})_{n ∈ N} has a lower bound *m* (*m* > 0)

then **Σu**_{n} diverges.

Particularly: if you can find α so that **n**^{α}u_{n} has a **finite limit λ**, then:

*Note: according to my textbooks, this condition is ***sometimes** called "Riemann Condition", but I get a feeling, not everybody has been able to agree on that... so please don't pop a coronary if your textbook decided to call it "Condition II.b" or something... The logic behind the naming goes with the fact that you are essentially bringing the problem back to a comparison with a Riemann Sum.

*Note 2: unperson points out there is something entirely different in complex analysis called the Cauchy-Riemann condition.*