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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?

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In order to do what? –  Qiaochu Yuan Jul 2 '11 at 23:33
    
@Qiaochu Yuan: Just for the sake of seeing whether it is bounded above or below. –  Damien Jul 2 '11 at 23:35
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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. –  mac Jul 2 '11 at 23:40
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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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