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Thread: Another binomial coeffeciant SUMMATION

  1. #1 Another binomial coeffeciant SUMMATION 
    Forum Ph.D. Heinsbergrelatz's Avatar
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    I believe IMO that summing binomial coefficients is very tricky. there are methods of induction to prove the result, but i think finding a way to come to actual result instead of M.induction is even better.



    how to yield this result?? any help would be appreciated. Umm... btw i know the way of induction and yea i have proved it that way, but i want some ways in actually retrieving the result(say if the answer wasn't given).

    thank you


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  3. #2  
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    I haven't tried it, but the following may work.

    Let f(x)=C0x^n - C1x^(n+1) + .... (-1)^n Cnx^2n.

    Get a closed form expression for f(x) and then integrate from 0 to 1.


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  4. #3 Re: Another binomial coeffeciant SUMMATION 
    . DrRocket's Avatar
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    Quote Originally Posted by Heinsbergrelatz
    I believe IMO that summing binomial coefficients is very tricky. there are methods of induction to prove the result, but i think finding a way to come to actual result instead of M.induction is even better.



    how to yield this result?? any help would be appreciated. Umm... btw i know the way of induction and yea i have proved it that way, but i want some ways in actually retrieving the result(say if the answer wasn't given).

    thank you



    makes no sense. What is the denominator in each term ?
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  5. #4 Re: Another binomial coeffeciant SUMMATION 
    Forum Ph.D. Heinsbergrelatz's Avatar
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by Heinsbergrelatz
    I believe IMO that summing binomial coefficients is very tricky. there are methods of induction to prove the result, but i think finding a way to come to actual result instead of M.induction is even better.



    how to yield this result?? any help would be appreciated. Umm... btw i know the way of induction and yea i have proved it that way, but i want some ways in actually retrieving the result(say if the answer wasn't given).

    thank you



    makes no sense. What is the denominator in each term ?

    my bad. Let me edit the question
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  6. #5  
    Forum Ph.D. Heinsbergrelatz's Avatar
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    i solved this just last night. okay imo i am weak at dealing with integrals involving combinations, and binomial expansions, so i will pose my working up and if any mistakes are seen please do correct.

    okay so from the above question i posed, it leads to this;



    then integrate from 1 to 0, like mathman said, then i will get the question i posed. now solving the actual integral is daunting.



    bt i realized the main trick behind this whole question was realizing a similar integral and solving it from there.










    now bck to the 2nd integral again.










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