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  • 1 Post By emakarov

Math Help - limit

  1. #1
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    limit

    x belongs to [0,1]

    limit n->infinity (limit m-> infinity cos 2n (m! pi x) )=?

    its value is 0 if x is irrational
    and it takes 1 if x is rational.

    please explain how it happens.
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  2. #2
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    Re: limit

    I am not sure about this nested limit, but we get the Dirichlet function if we swap limits:

    \lim_{m\to\infty}\left(\lim_{n\to\infty}\left(\cos  (m!\pi x)^{2n}\right)\right).

    If x is rational, then m!x is eventually an (even) integer. If x is irrational, then \cos(m!\pi x)\in(-1,1), so the inner limit is 0.
    Thanks from saravananbs
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