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Suppose that ${a_k}$ is a real valued increasing sequence such that

$$ \sum_{k=1}^{\infty} \frac{1}{a_k} = +\infty ,$$

i.e. $\{a_1,a_2,\ldots\}$ is a large set.

If $\lim a_{k+1} - a_{k} = \infty$, what can be said about the rate of growth of $a_{k+1} - a_{k}$? It is true that

$$ \lim \frac{k}{a_{k+1} - a_{k}} = \infty ?$$

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No, this is not true take $a_k =\ln k^k $

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  • $\begingroup$ Thanks for the clever example. Do you think there is also an example of this kind of sequence that also has gaps that grow faster than exponential, i.e. $\lim \frac{a_{k+1} - a_{k}}{e^{k}} = \infty$? $\endgroup$
    – shamisen
    Commented Feb 12, 2016 at 19:55
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    $\begingroup$ @shamisen No, because then it would converge by comparison tests. $\endgroup$ Commented Oct 14, 2017 at 17:20

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