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Seperating a Summation problem.

  1. May 3, 2008 #1
    [SOLVED] Seperating a Summation problem.

    1. The problem statement, all variables and given/known data
    The Problem:
    Separate a sum into 2 pieces (part of a proof problem).

    Using: [tex]X=
    \sum^{n}_{k=1}\frac{n!}{(n-k)!}
    [/tex]

    Solve in relation to n and X:
    [tex]
    \sum^{n+1}_{k=1}\frac{(n+1)!}{(n+1-k)!}
    [/tex]

    2. Relevant equations
    ???

    3. The attempt at a solution
    [tex]
    \sum^{n}_{k=1}[\frac{(n+1)!}{(n+1-k)!}]+\frac{(n+1)!}{(n+1-[n+1])!}
    [/tex]


    [tex]
    \sum^{n}_{k=1}[\frac{(n)!}{(n-k)!}*\frac{(n+1)}{(n+1-k)}]+\frac{(n+1)!}{(n+1-[n+1])!}
    [/tex]


    [tex]
    (n+1)*\sum^{n}_{k=1}[\frac{(n)!}{(n-k)!}*\frac{1}{(n+1-k)}]+(n+1)!}
    [/tex]


    I think this is fairly close but, I have no way of getting rid of the 1/(n+1-k) term.
     
  2. jcsd
  3. May 4, 2008 #2

    Defennder

    User Avatar
    Homework Helper

    Can you show that [tex]\sum_{k=2}^n \frac{n!}{(n-k+1)!} \ + \ \frac{n!}{(n+1-(n+1))!} = \sum^{n}_{k=1}\frac{n!}{(n-k)!} [/tex]?

    If you do that, you can express [tex]\sum^{n+1}_{k=1}\frac{(n+1)!}{(n+1-k)!}[/tex] as a summation starting from k=2. Then you should be able to get the desired expression.

    Try it out for some values of k and n, then you'll see a pattern.

    In general the pattern is [tex]\sum_{k=1}^{n}f(k) = \sum_{k=2}^{n} f(k-1)\ +\ f(n)[/tex]
     
  4. May 4, 2008 #3
    That helps quite a bit.

    [tex]
    n!+\sum^{n}_{k=2}\frac{n!}{(n+1-k)!}=\sum^{n}_{k=1}\frac{n!}{(n-k)!}=X
    [/tex]

    So continuing from this step:
    [tex]
    (n+1)*\sum^{n}_{k=1}\frac{n!}{(n-k+1)!}+\frac{(n+1)!}{(n+1-[n+1])!}
    [/tex]

    changing the Index and adding/subtracting n!
    [tex]
    (n+1)*(1-n!+n!+\sum^{n}_{k=2}\frac{n!}{(n-k+1)!})+(n+1)!
    [/tex]

    Solves the Equation in terms of n and X:
    [tex]
    (n+1)*(1-n!+X)+(n+1)!
    [/tex]

    Yep, that worked, now I can complete the rest of the proof :) Thank you very much. How to mark this "[solved]?"
     
  5. May 4, 2008 #4

    Defennder

    User Avatar
    Homework Helper

    Go to your first post in this thread, at the top right hand corner of the post marked "Thread Tools"
     
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