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Math Help - Product of these sums is a sum of the same form...

  1. #1
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    Product of these sums is a sum of the same form...

    Hi

    This time I'd like to know how to show this:

    Product of these sums is a sum of the same form...-asd.png


    Thanks for the help.


    I edit: This is my first attempt of proof...but I'm not too happy with it...it seems to work, but it's too vague...

    Product of these sums is a sum of the same form...-asd.png
    Attached Thumbnails Attached Thumbnails Product of these sums is a sum of the same form...-asd.png   Product of these sums is a sum of the same form...-asd.png  
    Last edited by Leviathantheesper; June 30th 2013 at 05:13 PM. Reason: Correcting...the property doesn't hold...but if we go to infinite it holds...
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  2. #2
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    Re: Product of these sums is a sum of the same form...

    Hey Leviathantheesper.

    Is y a constant for all k or does it change for a specific k?
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  3. #3
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    Re: Product of these sums is a sum of the same form...

    The way you have written it, with y constant, it is not true. (Hence chiro's question.) Rather you need something like
    \sum_{k=0}^n a_kx^k \sum_{k= 0}^n b_kx^k= \sum_{k= 0}^{2n} y_kx^k

    That is, you need that "y" different for each power of x and, obviously, if you are multiplying two nth degree polynomials together, you do NOT get an nthe degree polynomial, you get a 2n degree polynomial:

    (a+ bx)(c+ dx)= ac+ (ad+ bc)x+ (bd)x^2.
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  4. #4
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    Re: Product of these sums is a sum of the same form...

    Quote Originally Posted by HallsofIvy View Post
    The way you have written it, with y constant, it is not true. (Hence chiro's question.) Rather you need something like
    \sum_{k=0}^n a_kx^k \sum_{k= 0}^n b_kx^k= \sum_{k= 0}^{2n} y_kx^k

    That is, you need that "y" different for each power of x and, obviously, if you are multiplying two nth degree polynomials together, you do NOT get an nthe degree polynomial, you get a 2n degree polynomial:

    (a+ bx)(c+ dx)= ac+ (ad+ bc)x+ (bd)x^2.
    Yes, y is not a constant, that was my mistake.

    And...I know i don't get a nth degree polynomial... I edited some time ago to change the picture and replace the n with infinite (I replaced it yesterday)...
    And the only thing I have to prove is that the result is a sum of the same form.
    This, correcting the y with y_k
    Last edited by Leviathantheesper; July 1st 2013 at 09:59 AM.
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  5. #5
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    Re: Product of these sums is a sum of the same form...

    Again, that certainly is NOT true! Unless you mean, as both chiro and I said before, you mean \sum_{k=0}^\infty y_kx^k.

    And, in that case, it is pretty close to trivial. The product of any two terms, a_ix^i times b_jx^j must be of the form a_ib_jx^{j+i} so the only thing you can have are "numbers times non-negative powers of x" and that is all the term " \sum_{k=0}^\infty y_kx^k means!
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  6. #6
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    Re: Product of these sums is a sum of the same form...

    Quote Originally Posted by HallsofIvy View Post
    Again, that certainly is NOT true! Unless you mean, as both chiro and I said before, you mean \sum_{k=0}^\infty y_kx^k.

    And, in that case, it is pretty close to trivial. The product of any two terms, a_ix^i times b_jx^j must be of the form a_ib_jx^{j+i} so the only thing you can have are "numbers times non-negative powers of x" and that is all the term " \sum_{k=0}^\infty y_kx^k means!

    I have just said:
    This, correcting the y with y_k

    And thanks, that's it...
    Last edited by Leviathantheesper; July 1st 2013 at 11:14 AM.
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