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Math Help - Special sequence

  1. #1
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    Special sequence

    Let \{a_n\}_{n=1}^{\infty}\subset N be strictly monotone increasing sequence

    \displaystyle<br />
\varphi(n)=\,\text{card}\,\{k|\,a_k\leq n\} and \varphi(n)\thicksim\frac n{\ln(n)}\ ,\ n\to\infty

    Prove that

    \sum_{n=1}^\infty \frac 1{a_{a_n}}<+\infty,\ \text{and}\ \,\sum_{n=1}^\infty \frac 1{a_n}=+\infty<br />
    Last edited by ns1954; January 21st 2010 at 11:02 AM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by ns1954 View Post
    Let \{a_n\}_{n=1}^{\infty}\subset N be strictly monotone increasing sequence

    \displaystyle<br />
\varphi(n)=\,\text{card}\,\{k|\,a_k\leq n\} and \varphi(n)\thicksim\frac n{\ln(n)}\ ,\ n\to\infty

    Prove that

    \sum_{n=1}^\infty \frac 1{a_{a_n}}<+\infty,\ \text{and}\ \,\sum_{n=1}^\infty \frac 1{a_n}=+\infty<br />
    Are these challenge questions? Are you just going to pose forty-seven million challenge questions without A) specifying whether they are indeed challenge questions B) contributing to MHF?
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