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Telescoping series convergence question

  1. Jun 20, 2015 #1
    1. The problem statement, all variables and given/known data

    Hello, this problem is from a well-known calc text:

    Σ(n=1 to ∞) 8/(n(n+2)


    2. Relevant equations

    What I have here is decomposingg the problem into Σ(n=1 to ∞)(8/n -(8/n+2)


    3. The attempt at a solution

    I have the series sum as equaling (8/1-8/3) + (8/2-8/4) + (8/3-8/5) + (8/4-8/6) + … + (8/n-(8/n+2)

    So, I have everything canceling but 8/1, 8/2, and 8/n+2. The last arm reaches 0 as n approaches ∞, so I have 8/1 + 8/2 = 12.

    However, I think I am missing something here, the answer in the back of the book is not 12.

    Thank you all for your help in advance.

    SY
     
  2. jcsd
  3. Jun 20, 2015 #2

    SammyS

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    Check your algebra.

    ##\displaystyle \ \frac{8}{n}-\frac{8}{n+2}=\frac{8n+16-8n}{n(n+2)} \ ##

    ##\displaystyle \ \quad \quad \ne\frac{8}{n(n+2)} \ ##​

    Also, be sure to use an adequate amount of parentheses.
     
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