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Sequences Test Question

  1. Jun 30, 2009 #1
    Show that ([tex]^{n}_{n}[/tex]) - ([tex]^{n}_{n-1}[/tex]) + ([tex]^{n}_{n-2}[/tex]) - ([tex]^{n}_{n-3}[/tex]) + ...([tex]^{n}_{0}[/tex]) = 0

    (a+b)[tex]^{n}[/tex] = [tex]\sum^{\infty}_{\nu=0}[/tex] ([tex]^{n}_{\nu}[/tex])a[tex]^{\nu}[/tex]b[tex]^{n-\nu}[/tex]


    a=1
    b=-1

    0 = (1+(-1))[tex]^{n}[/tex] = [tex]\sum^{\infty}_{\nu=0}[/tex]([tex]^{n}_{\nu}[/tex]) 1[tex]^{\nu}[/tex](-1)[tex]^{n-\nu}[/tex] = [tex]\sum^{\infty}_{\nu=0}[/tex]([tex]^{n}_{\nu}[/tex])(-1)[tex]^{n-\nu}[/tex] =

    ...I don't know what to do here...

    = [tex]\sum^{n}_{\nu=0}[/tex]([tex]^{n}_{n-\nu}[/tex])(-1)[tex]^{\nu}[/tex] = ([tex]^{n}_{n}[/tex]) - ([tex]^{n}_{n-1}[/tex]) + ([tex]^{n}_{n-2}[/tex]) - ([tex]^{n}_{n-3}[/tex]) + ...([tex]^{n}_{0}[/tex])


    That is if that last equality is correct and makes sense to be there.

    Any suggestions?

    Also, how can I make it so everything that follows the Sigma doesn't look like it is being superscripted?

    Thanks!
     
    Last edited: Jun 30, 2009
  2. jcsd
  3. Jun 30, 2009 #2
    a=1, b=-1
    (1+(-1))n=0

    Now try a=-1, b=1 so that

    (-1+1)n=0

    Or separate the odds from the even and create two separate sums:
    sum (even) - sum (odd) = 0
     
  4. Jun 30, 2009 #3
    Does this [tex]\sum[/tex][tex]^{\infty}_{\nu=0}[/tex] ([tex]^{n}_{\nu}[/tex]) imply [tex]\sum[/tex][tex]^{n}_{\nu=0}[/tex] ([tex]^{n}_{\nu}[/tex]) because it's for every n choose [tex]\nu[/tex] so there can only be n many [tex]\nu[/tex]s?
     
  5. Jun 30, 2009 #4
    I told you to use something like this:
    [tex]\sum_{v=0}^{n/2}\binom{n}{n-2v}(-1)^{2v}*1^{n-2v}+\sum_{v=0}^{n/2}\binom{n}{n-2v-1}(-1)^{2v-1}*1^{n-2v-1}[/tex]

    For the both sums a=-1 and b=1, so that 0-0=0

    Use \binom{ } { } for binomial coefficients.

    Regards.
     
    Last edited: Jun 30, 2009
  6. Jun 30, 2009 #5
    A more fundamental question I have, I guess, is if I know that [tex]\sum[/tex][tex]^{\infty}_{\nu=0}[/tex] ([tex]^{n}_{\nu}[/tex]) a[tex]^{\nu}[/tex] b[tex]^{n-\nu}[/tex] = (a+b)[tex]^{n}[/tex] then do I know that
    [tex]\sum[/tex][tex]^{n}_{\nu=0}[/tex] ([tex]^{n}_{n-\nu}[/tex]) a[tex]^{\nu}[/tex] b[tex]^{n-\nu}[/tex] = (a+b)[tex]^{n}[/tex]? How does the changing of infinity to n and the [tex]\nu[/tex] to n - [tex]\nu[/tex] impact the series?
     
    Last edited: Jun 30, 2009
  7. Jun 30, 2009 #6

    Dick

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    Science Advisor
    Homework Helper

    n is an integer, right? Then the generalized binomial coefficient C(n,nu) of n and nu is zero for nu>n, also right? There's a zero factor in the definition. That's how you truncate the infinite sum. Though I'm not sure why you have an infinite sum to begin with. And in that case, sure, C(n,nu)=C(n,n-nu).
     
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