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Thread: Convergency of a sequence

  1. #1
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    Convergency of a sequence

    Hi,
    Let a(n) be a sequence such that lim(a(2n)-a(n))=0.
    Prove or disprove:the sequence a(n) converges.
    Thank's in advance.
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  2. #2
    Junior Member Walagaster's Avatar
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    Re: Convergency of a sequence

    Quote Originally Posted by hedi View Post
    Hi,
    Let a(n) be a sequence such that lim(a(2n)-a(n))=0.
    Prove or disprove:the sequence a(n) converges.
    Thank's in advance.
    Have you had any luck with this? What have you tried? I would suggest trying to make a counterexample.
    Thanks from topsquark
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  3. #3
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    Re: Convergency of a sequence

    Quote Originally Posted by hedi View Post
    Hi,
    Let a(n) be a sequence such that lim(a(2n)-a(n))=0.
    Prove or disprove:the sequence a(n) converges.
    Thank's in advance.
    Let $\displaystyle n = \prod_{p\text{ is prime}} p^{b_p}$ be the prime factorization of $n$.

    Then let
    $\displaystyle a_n = \prod_{\begin{matrix} p\text{ is prime} \\ p \neq 2\end{matrix}}p^{b_p}$

    So, here is a quick table of values so you can see the pattern:

    $\displaystyle \begin{matrix}n & a_n \\ \hline 1 & 1 \\ 2 & 1 \\ 3 & 3 \\ 4 & 1 \\ 5 & 5 \\ 6 & 3 \\ 7 & 7 \\ 8 & 1 \\ 9 & 9 \\ 10 & 5 \\ 11 & 11 \\ 12 & 3\end{matrix}$

    Here it is easy to see that for all $n$, $a_{2n} = a_n$. But, this sequence quite clearly does not converge.
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