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Math Help - [SOLVED] Positive and divergent infinite sequence

  1. #1
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    [SOLVED] Positive and divergent infinite sequence

    I need to construct a divergent sequence \{ a_n \} such that a_n>0, \forall n \in \mathbb{N}, and \lim \limits_{n \to \infty } \frac{a_{n+1}}{a_n}=1

    I can't think of one
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    Moo
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    Hello,
    Quote Originally Posted by akolman View Post
    I need to construct a divergent sequence \{ a_n \} such that a_n>0, \forall n \in \mathbb{N}, and \lim \limits_{n \to \infty } \frac{a_{n+1}}{a_n}=1

    I can't think of one
    a_n=\frac 1n ?


    Note that if you apply the ratio test and find the limit to be 1, the test is inconclusive.
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    Quote Originally Posted by Moo View Post
    Hello,

    a_n=\frac 1n ?


    Note that if you apply the ratio test and find the limit to be 1, the test is inconclusive.
    The sequence a_n=\frac 1n converges to 0, because \lim \limits_{n \to \infty} a_n=\lim \limits_{n \to \infty} \frac{1}{n}=0.

    That will work for the Harmonic series \sum ^\infty _{n=1} \frac{1}{n}, but I don't think that will work in this case.

    I need something like \lim \limits_{n \to \infty} a_n=\infty and \lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n}=1
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  4. #4
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    Quote Originally Posted by akolman View Post
    The sequence a_n=\frac 1n converges to 0, because \lim \limits_{n \to \infty} a_n=\lim \limits_{n \to \infty} \frac{1}{n}=0.

    That will work for the Harmonic series \sum ^\infty _{n=1} \frac{1}{n}, but I don't think that will work in this case.
    Please excuse me, I thought we were working on series

    I need something like \lim \limits_{n \to \infty} a_n=\infty and \lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n}=1
    So take a_n=n

    \frac{a_{n+1}}{a_n}=\frac{n+1}{n}=1+\frac 1n which goes to 1 as n goes to infinity.
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