Riemann condition

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).

1. If you can find α > 1 so that:
   (n^αu_n)_{n ∈ N} has an upper bound
   then Σu_n converges.

2. If you can find α ≤ 1 so that:
   (n^αu_n)_{n ∈ N} has a lower bound m (m > 0)
   then Σu_n diverges.

3. Particularly: if you can find α so that n^αu_n has a finite limit λ, then:

   • (λ > 0 and α ≤ 1) ⇒ Σu_n diverges
   • (α < 1) ⇒ Σu_n converges

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.