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Limit of a series

  1. Feb 6, 2008 #1
    1. The problem statement, all variables and given/known data
    Determine whether the series is convergent or divergent. If convergent, find its sum.

    sum of [1+(2)^n] / (3^n)
    from 1 to inf


    2. Relevant equations
    I know that the sum of a geometric series is 1/(1-r)


    3. The attempt at a solution
    The sum of a series is the limit of its partial sums.
    I separate the summation into 2 parts: 1/(3^n) + (2^n)/(3^n)

    I can see from this that the limits of both of these approach 0, so I conclude that the sum the series is 0.

    However, my book says the answer is 5/2 and I tried to solve this a different way and got 5/2 as well. I re-wrote the separate summations as (1/3)^n + (2/3)^n and notice the ratio, r, is 1/3 and 2/3, respectively. Applying the "relevant equation" of 1/(1-r) I solve the summations and get 5/2.

    However, if the sum of a series is the limit of its partial sums, why am I getting a different value for my first attempt?
     
  2. jcsd
  3. Feb 6, 2008 #2

    Dick

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    You are confusing the limit of a sequence with the sum of a series. That's all.
     
  4. Feb 6, 2008 #3
    My book says that "the sum of a series is the limit of the sequence of partial sums."
     
  5. Feb 6, 2008 #4

    Dick

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    You didn't do any partial sums. You just looked at the limits of the individual terms.
     
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