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Homework Help: Calculate sums math problem

  1. Dec 10, 2005 #1

    daniel_i_l

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    Gold Member

    I need to calculate
    [tex]
    \sum_{i=1}^n \frac{1}{i(i+1)}
    [/tex]
    useing the fact that:
    [tex]
    \sum_{i=1}^n F(i) - F(i-1) = F(n) - F(0)
    [/tex]
    now I chose the function
    [tex]
    F(i) = \frac{1}{i} \frac{1}{(i+1)} ... \frac{1}{(i+r)}
    [/tex]
    so
    [tex]
    F(i)-F(i-1)=(\frac{1}{i}\frac{1}{(i+1)} ... \frac{1}{(i+r-1)})(\frac{1}{(i+r)}-\frac{1}{(i-1)})
    [/tex]
    now I want to use that to calcualte the sum chooseing r as 2, but I'm stuck because the F(0) is undefined, and because of the
    [tex]
    \frac{1}{i+r}-\frac{1}{i-1})
    [/tex]
     
    Last edited: Dec 10, 2005
  2. jcsd
  3. Dec 10, 2005 #2

    shmoe

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    Science Advisor
    Homework Helper

    You need to choose F(i) so that

    [tex]
    F(i)-F(i-1)=\frac{1}{i(i+1)}
    [/tex]

    Your choice of F(i) does not satisfy this (what is that r anyways?) Try another choice of F, with a hint-think partial fractions.
     
  4. Dec 10, 2005 #3

    Physics Monkey

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    Science Advisor
    Homework Helper

    Why on earth would you choose that three term nightmare for F? There is a much easier way to do it. Hint: what is [tex] \frac{1}{i+1} - \frac{1}{i} [/tex]?
     
  5. Dec 10, 2005 #4

    daniel_i_l

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    Gold Member

    Thanks a lot!
     
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