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Evaluating a Series

  1. Nov 7, 2011 #1
    1. The problem statement, all variables and given/known data
    First, I'd like to thank everybody a head of time. You guys have been an enormous help.

    Second, I don't mind telling you that I'm finding sequences and series extremely frustrating. I usually pick this stuff up like nobody's business.

    My problem is attached, but I will copy it down as I understand it.


    [itex]
    r = \frac{11}{24}
    [/itex]

    [itex]
    \sum _{i=1} ^\inf nr^n
    [/itex]

    Mysteriously, this can be rewritten as
    [itex]
    \sum_{i=1} ^n ir^i = \frac{ nr^{n+2} - (n+1)r^{n+1} + r }{ (1 - r)^2 }
    [/itex]


    2. Relevant equations
    [itex]
    \sum _{i=1} ^\inf nr^n \rightarrow
    n \sum _{i=1} ^\inf r^n \rightarrow
    n \frac{1}{1-r}
    [/itex]


    3. The attempt at a solution

    [itex]
    \sum _{i=1} ^\inf nr^n \rightarrow
    n \sum _{i=1} ^\inf r^n \rightarrow
    n \frac{1}{1-r}
    [/itex]

    [itex]
    \frac{1}{1-r}
    [/itex]
    This is a number greater than one,
    [itex]
    \frac{24}{13}
    [/itex]

    So as n goes to infinity, the number just gets bigger and bigger right? Diverges to infinite is, apparently, not the answer.
     

    Attached Files:

  2. jcsd
  3. Nov 7, 2011 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You switched summation indices incorrectly. The series is either sum_{i} i*r^i or sum_{n} n*r^n; it is NOT sum_{i} n*r^n or whatever. Go back and read the question more carefully.
    Remember: |r| = 11/24 is less than 1---that matters a lot.

    RGV
     
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