Results 1 to 4 of 4

Math Help - limits in double sum

  1. #1
    Newbie
    Joined
    Jul 2011
    Posts
    9

    limits in double sum

    Could you explain me the change in the limits of this double sum?
    \sum_{a=0}^n\binom{n}{a}(1+x(1-y))^{n+a}x^a(y+x(1-y))^a\sum_{b=0}^{n+a}\binom{n+a}{b}k^b=

    =\sum_{b=0}^{2n}\sum_{a=max(0,b-n)}^n\binom{n}{a}\binom{n+a}{b}(1+x(1-y))^{n+a}x^a(y+x(1-y))^ak^b

    Is there any property about this change and how can I understand the correct change in a similar case?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: limits in double sum

    Denote by f(a,b) what is summed. We have to show that \sum_{a=0}^n\sum_{b=0}^{n+a} f(a,b) =\sum_{b=0}^{2n}\sum_{a=\max (0,b-n)}^nf(a,b). We have

    \begin{align*}\sum_{b=0}^{2n}\sum_{a=\max (0,b-n)}^nf(a,b) &=\sum_{b=0}^{n}\sum_{a=\max (0,b-n)}^nf(a,b)+\sum_{b=n+1}^{2n}\sum_{a=\max (0,b-n)}^nf(a,b) \\&= \sum_{b=0}^{n}\sum_{a=0}^nf(a,b)+\sum_{b=n+1}^{2n}  \sum_{a=b-n}^nf(a,b)\end{align*}
    and \sum_{a=0}^n\sum_{b=0}^{n+a} f(a,b) =\sum_{a=0}^n\sum_{b=0}^n f(a,b)+\sum_{a=0}^n\sum_{b=n+1}^{n+a} f(a,b). Hence you only have to show that \sum_{b=n+1}^{2n}\sum_{a=b-n}^nf(a,b) =\sum_{a=0}^n\sum_{b=n+1}^{n+a} f(a,b). To see that, we notice that the set in which a and b lie is a triangle.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jul 2011
    Posts
    9

    Re: limits in double sum

    Quote Originally Posted by girdav View Post
    ... we notice that the set in which a and b lie is a triangle.
    Could you explain me what you mean?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: limits in double sum

    For the first sum, we sum for (a,b) = (1,n+1),\ldots (n,n+1),  (2,2+n),\ldots, (n,n+2),\ldots, (n,2n) and if your draw these points on the upper half plane you will get a triangle.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. limits in double sum
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: July 27th 2011, 09:27 AM
  2. limits of double integration
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 11th 2009, 03:50 PM
  3. finding the limits of double integral
    Posted in the Calculus Forum
    Replies: 6
    Last Post: October 10th 2009, 08:23 PM
  4. Double Integral Limits
    Posted in the Calculus Forum
    Replies: 2
    Last Post: June 1st 2009, 04:47 PM
  5. Double integration: limits
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 8th 2008, 07:28 PM

Search Tags


/mathhelpforum @mathhelpforum