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Thread: pi is the limit of what function?

  1. #1 pi is the limit of what function? 
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    for what function that isn't trigonometric is pi the limit? as in it looks like your basic function?

    pi itself cant be in the function. i found this but i'm not sure what it means

    2 to the power of 4n divided by n times 2n over n squared??? its all jumbled up


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  3. #2  
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    It represents binomial coefficients.
    Binomial theorem - Wikipedia, the free encyclopedia


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  4. #3  
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    Quote Originally Posted by Harold14370 View Post
    It represents binomial coefficients.
    Binomial theorem - Wikipedia, the free encyclopedia
    ok thank you. i thought binomial[2n, n] was 2n divided by n missing the line of division.
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  5. #4  
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    There is no formula for pi, just ones people make up in an attempt to compute it.





    Edit: pi = c/d, circumference divided by the diameter, so that's a formula for pi. But what I mean is, there is no formula where you can go and compute the value of pi to as many places as you want. People have introduced different formulas for pi and used them for the purpose of calculating pi, but they 'made up' the formula themselves.
    Last edited by Ledger; November 23rd, 2011 at 03:29 PM.
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  6. #5  
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    ok i found several infinite series in where pi is the limit. nothing on your basic function though, there probably is no function for pi.
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  7. #6  
    Forum Ph.D. merumario's Avatar
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    Pi has been able to be computed that it is of 1.2trillion decimal places,which does not end at zero...mathematicans have used several expression with real numbers.and note: pi is an irrational number.
    "I am sorry for making this letter longer than usual.I actually lacked the time to make it shorter."###
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  8. #7  
    ***** Participant Write4U's Avatar
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    Perhaps it is an irrational number because it is not a function, it is a constant. Similar to 'c" (sol) being an irrational speed but self evidently true.
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  9. #8  
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    Pi is a ratio which happens to be a transcendental number - it is not algebraic.

    It's best described by sticking to the description as a transcendental number, even though it's irrational, because not all irrational numbers are transcendental. Transcendental number - Wikipedia, the free encyclopedia
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  10. #9  
    Forum Ph.D. merumario's Avatar
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    Yeah its more like c, its a constant...since the ratio of the circumference to its radius is always true for any circle and shpere...which is ~3.14159256.
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    Quote Originally Posted by brody View Post
    for what function that isn't trigonometric is pi the limit?
    Well not even that, actually.

    Hmm......there may be other ways of thinking about this - here;s one. Assume that is transcendental (the proof that it is transcendental is ferocious, so just assume it!)

    1.Think what it means for a real number to be transcendental (Hint it involves polynomials with integer coefficients)

    2. Think how polynomial functions are written (Hint it involves summation)

    3. Think about the relation between series and sequences (Hint it involves partial sums)

    4. Think how this may be applicable - if at all - to a polynomial function

    5. Think why one may conclude that cannot be the limit of ANY function

    Note the word THINK in all of the above
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  12. #11  
    Forum Ph.D. merumario's Avatar
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    nice one,just think an d you it clear from another perspective.
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  13. #12  
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    No in function: Check
    No (real-valued *cough cough*) trigonometry: Check
    Dis muthufukka go hard. -Quote
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  14. #13  
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    Oops! I just checked and I was wrong - is the limit of numerous non-trig functions.

    Worse, I expressed myself rather arrogantly. My humble apologies.

    *blush*
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    Some 22 centuries ago, Archimedes computed approximated values of considering regular polygons and their circumscribed and inscribed circles. We can try this again :
    Let
    the radius of the circumscribed circle,
    the side of a regular polygon with n sides.
    the side of a regular polygon with 2n sides.
    Then


    Using this, we can build by induction two functions S and f :




    f(n) tends toward when n tends toward infinity.

    [Sorry, I had to edit this because my first transcription was false. My last 2012 mistake ]
    Last edited by caKus; January 1st, 2013 at 02:45 AM. Reason: Transcription error
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  16. #15  
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    Quote Originally Posted by Guitarist View Post
    I expressed myself rather arrogantly.
    You're much too harsh on yourself. I thought your statements were arrogant in the least... and possibly still correct, but I'm not qualified to verify that.

    Wolfram Research provides several limit representations for . And each of them seems to involve either the gamma function, infinite summation/production, or both.

    But nothing as far as one's basic polynomial (or rational) function, like you stated. Which interests me... Is there such a proof that is not the limit of any polynomial or rational function? Surely this is closely tied with 's transcendence?

    Assume that is transcendental (the proof that it is transcendental is ferocious...
    Would you be referring to the Lindermann-Weierstrass theorem?
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  17. #16  
    Forum Junior epidecus's Avatar
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    By the way, was just a derivation of an identity for the inverse sine function.

    It's all but "non-trigonometric", since for all complex . This follows the identity .

    (note that here , where the former is more common to pure mathematics and the latter to engineering)

    So all that was needed to be done was create an infinite limit such that the variable can be disregarded and the expression remains.

    This is anything but unique since it branches off a trivial identity. But the limit is still a surprising one.
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