I have a rather general question about sequences:

How do you determine whether it is appropriate to find an upper or lower bound of a sequence? In particular, suppose $a_n = \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{p_n}$ where $p_n$ denotes the $n^{th}$ prime. How do I know whether to bound the sequence from above or below?

  • $\begingroup$ In order to do what? $\endgroup$ Jul 2, 2011 at 23:33
  • $\begingroup$ @Qiaochu Yuan: Just for the sake of seeing whether it is bounded above or below. $\endgroup$
    – Damien
    Jul 2, 2011 at 23:35
  • 4
    $\begingroup$ Isn't your question a tautology then? To prove that it's bounded above, bound it from above; to prove that it's not bounded above, bound it from below (by some sequence you know to not be bounded above). Similarly if you want to prove/disprove that it's bounded below. $\endgroup$
    – mac
    Jul 2, 2011 at 23:40

1 Answer 1


Your question is vague, but for your particular example, the sequence a_n is bounded below by zero and doesn't have an upper bound. Look at here for more: https://secure.wikimedia.org/wikipedia/en/wiki/Proof_that_the_sum_of_the_reciprocals_of_the_primes_diverges


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