Results 1 to 4 of 4

Thread: Iterated sums

  1. #1
    Junior Member
    Joined
    Jun 2009
    Posts
    33

    Iterated sums

    Sorry if this is the wrong forum.

    Is there a "nice" formula for the following? :

    $\displaystyle \sum_{k_2=0}^{k_1}\sum_{k_3=0}^{k_2}\sum_{k_4=0}^{ k_3} ... \sum_{k_n=0}^{k_{n-1}}1$

    I do the first few iterations and it just seems to get messy :S

    Thanks for any help
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,743
    Thanks
    2814
    Awards
    1
    Quote Originally Posted by Aileys. View Post
    Sorry if this is the wrong forum.

    Is there a "nice" formula for the following? :

    $\displaystyle \sum_{k_2=0}^{k_1}\sum_{k_3=0}^{k_2}\sum_{k_4=0}^{ k_3} ... \sum_{k_n=0}^{k_{n-1}}1$
    You are using the same indices more than once.
    Do you mean
    $\displaystyle \sum_{j_2=0}^{k_1}\sum_{j_3=0}^{k_2}\sum_{j_4=0}^{ k_3} ... \sum_{j_n=0}^{k_{n-1}}1$?

    If not how can the indices change like that?
    What am I missinng?

    Edit
    I will say that $\displaystyle \sum_{j_2=0}^{k_1}\sum_{j_3=0}^{k_2}\sum_{j_4=0}^{ k_3} ... \sum_{j_n=0}^{k_{n-1}}1$$\displaystyle =\prod\limits_{j = 1}^{k_{n - 1} } {\left( {k_j + 1} \right)} $
    Last edited by Plato; Oct 14th 2009 at 03:57 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jun 2009
    Posts
    33
    Yes the indices are supposed top be like that, - that's why it's messy!

    For example $\displaystyle \sum_{k_2=0}^{k_1}\sum_{k_3=0}^{k_2}1 = \sum_{k_2=0}^{k_1}(k_2+1) = \frac{k_1(k_1+1)}{2}-(k_1+1)$

    Perhaps I'm abusing notation... if so, sorry! But hopefully that^ explains what I'm trying to do
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Mar 2008
    Posts
    934
    Thanks
    33
    Awards
    1
    Quote Originally Posted by Aileys. View Post
    Sorry if this is the wrong forum.

    Is there a "nice" formula for the following? :

    $\displaystyle \sum_{k_2=0}^{k_1}\sum_{k_3=0}^{k_2}\sum_{k_4=0}^{ k_3} ... \sum_{k_n=0}^{k_{n-1}}1$

    I do the first few iterations and it just seems to get messy :S

    Thanks for any help
    The sum is the same as the number of sequences
    $\displaystyle 0 \leq k_n \leq k_{n-1} \leq \dots \leq k_2 \leq k_1 $,

    which is the same as the number of multisets of size $\displaystyle n$ taken from a set of $\displaystyle k_1$ objects,

    which is
    $\displaystyle \binom{k_1 +n -1}{n}$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Iterated sup
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Mar 25th 2011, 03:49 PM
  2. Iterated integrals
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Oct 25th 2010, 07:23 PM
  3. iterated integral
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Jan 24th 2010, 12:26 AM
  4. Iterated Integrals
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Jan 15th 2010, 04:27 PM
  5. iterated integrals
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Jul 26th 2008, 03:19 PM

Search Tags


/mathhelpforum @mathhelpforum