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Question over a Series

  1. Apr 29, 2016 #1
    1. The problem statement, all variables and given/known data
    I was watching a PatrickJMT video on ratio test and he gave this problem. I solved it before he did, and he got that it was divergent. He didn't simplify it initially, so our methods of approach are different. Did I do something wrong? I checked my calculator to make sure all my simplifications were correct. So I'm assuming I did something near the limit portion that was incorrect (if at all). Thanks


    ∑ n/31+3n
    n=1

    2. Relevant equations


    3. The attempt at a solution

    ∑ n/31+3n = n/(3⋅27n)
    n=1

    lim |(n+1)/(3⋅27n+1) / (n/3 *27n)|
    n→∞

    = lim |(n+1)/n| * (1/27)
    n→∞
    = 1 * 1/27
    =1/27 < 1

    Convergent by Ratio Test
     
  2. jcsd
  3. Apr 29, 2016 #2

    Math_QED

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    Homework Helper

    Did you use the ratio test correctly?
     
  4. Apr 29, 2016 #3
    IMG_1809.jpg
    Nope. Got it, thanks.

    [edit]

    Actually I typed my work incorrectly. I'll fix it. but to be sure...
     
  5. Apr 29, 2016 #4

    Ray Vickson

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    Science Advisor
    Homework Helper

    It is convergent, as you have shown correctly using the ratio test.

    You can actually find the sum ##\sum_{n=1}^N n/3^{3n+1}## explicitly (as a reasonably simple closed-from expression in ##N##), then take the limit as ##N \to \infty##. All you need to do is find a formula for ##f_N(x) =\sum_{n=1}^N n x^n##, and these are widely available, or you can work it out as ##f_N(x) = x (d/dx) g_N(x)##, where ##g_N(x) = \sum_{n=1}^N x^n##, which is elementary.
     
    Last edited: Apr 29, 2016
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