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Math Help - topology ans sequences

  1. #1
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    topology ans sequences

    Can you find any sequence of rationals such that it converges to π/4?

    use the fact that the rationals are dense in the reals,
    or you can think about the decimal expansion of π/4. How can you get rational numbers out of that, rational numbers that converge to π/4?

    Can anybody help me with this immediately??
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  2. #2
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    Re: topology ans sequences

    use the fact that the rationals are dense in the reals to find the answer for this,,,

    Let f:R→R be a continuous function such that f(q)=sinq for q∈Q (rational numbers). Find the value of f(π/4).
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  3. #3
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    Re: topology ans sequences

    Quote Originally Posted by chath View Post
    ...
    or you can think about the decimal expansion of π/4
    If the question was just asking about \pi, I'd suggest this rational sequence to you:
    a_0 = 3
    a_1 = 3.1
    a_2 = 3.14
    a_3 = 3.141
    a_4 = 3.1415
    Etc.
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  4. #4
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    Re: topology ans sequences

    no not about pi.i need a rational function that converges to pi/4
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  5. #5
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    Re: topology ans sequences

    I have this idea...to find f(π/4) we need to pick a sequence xn→π/4 and investigate what f(xn) converges to....will that be ok for this???
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  6. #6
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    Re: topology ans sequences

    Quote Originally Posted by chath View Post
    no not about pi.i need a rational function that converges to pi/4
    The example I gave was meant to be suggestive of how you could construct a rational sequence for \frac{\pi}{4}.

    Think about it a minute. Each of those, each term in the sequence, is rational, and their limit goes to \pi.

    So can you think of a different-but-related sequence of rationals whose limit goes to \frac{\pi}{4} \ ?
    Thanks from chath
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  7. #7
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    Re: topology ans sequences

    Quote Originally Posted by chath View Post
    I have this idea...to find f(π/4) we need to pick a sequence xn→π/4 and investigate what f(xn) converges to....will that be ok for this???
    it sounds like you're thinking along these lines:

    Find a sequence b_n converging to \frac{\sqrt{2}}{2}, and then f^{-1}(b_n) will converge to \pi.

    If so, that isn't going to get you very far, because there's a complication with the function not having an inverse (though, by restricting domains, that can be delt with), and, vastly more intractable, you're going to need these f^{-1}(b_n) to be rational.
    Thanks from chath
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