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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 un positive (un≥ 0 for all n).

  1. If you can find α > 1 so that:
    (nαun)n ∈ N has an upper bound
    then Σun converges.

  2. If you can find α ≤ 1 so that:
    (nαun)n ∈ N has a lower bound m (m > 0)
    then Σun diverges.

  3. Particularly: if you can find α so that nαun 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.

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