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Math Help - Pi as summation

  1. #1
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    Pi as summation

    So I've seen that pi can be written as a series:

    http://upload.wikimedia.org/math/4/b...be1ac3be60.png

    pi = 4 * series(((-1)^k)/2k+1)

    So pi = 4/1 - 4/3 + 4/5 - 4/7 + ...

    Now I know that Q is an ordered field so if a and b are in Q, if I add them, the result a+b is another element in Q.

    But if I sum up infinitely many rational numbers I can get an irrational? Wondering how this works.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by anomaly View Post
    So I've seen that pi can be written as a series:

    http://upload.wikimedia.org/math/4/b...be1ac3be60.png

    pi = 4 * series(((-1)^k)/2k+1)

    So pi = 4/1 - 4/3 + 4/5 - 4/7 + ...

    Now I know that Q is an ordered field so if a and b are in Q, if I add them, the result a+b is another element in Q.

    But if I sum up infinitely many rational numbers I can get an irrational? Wondering how this works.
    Is this really that surprising? Isn't every irrational number the limit of a sequence of rational approximations, which can be represented as the infinite sum of rationa numbers.
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  3. #3
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    I know, but when you take the limit of a sum of a bunch of rationals you get an irrational, even though the rationals are closed under addition? That's what's strange about it.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by anomaly View Post
    I know, but when you take the limit of a sum of a bunch of rationals you get an irrational, even though the rationals are closed under addition? That's what's strange about it.
    Closed under finite addition.
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  5. #5
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    So when you take a limit of something you can converge to something completely different?
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by anomaly View Post
    So when you take a limit of something you can converge to something completely different?
    I have no real idea what that means, but I suppose the answer is yes.
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  7. #7
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    I mean having a summation converging to a value that is not within the original set.

    Ok, thank you.
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