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Calculus 2 - Infinite Series

  1. Oct 28, 2011 #1
    1. The problem statement, all variables and given/known data

    Find the sequence of partial sums {S_n} and evaluate the limit of {S_n} for the following series

    .9+.09+.009+...

    What is .9+.09+.009+... equal to?

    2. Relevant equations



    3. The attempt at a solution

    For the first part of the question (find the sequence of partial sums {S_n})
    S_n=9(1/10)^n where n >= 1
    my teachers assistant marked my answer correct

    for the second part of the question (evaluate the limit of {S_n} for the following series .9+.09+.009+...)
    I evaluated the limit of S_n by just simply taking the limit of S_n as n goes to infinity
    lim n->inf S_n = 9*lim n->inf (1/10)^n = 0
    My teachers assistant marked my question wrong and put
    S_n = sigma[1,4] 9(1/10)^ character
    I can't read what character he put
    I don't see how this answer is correct and my answer is wrong. If my answer to finding S_n is correct then why can't I just evaluate the limit as n goes to infinity of S_n to "evaluate the limit of {S_n}? I don't understand what's wrong with my work.

    for the third part (What is .9+.09+.009+... equal to?)
    .9+.09+.009+... = sigma[n=1,inf] (1/10)^n = 9* (1/10)/(1-1/10) = 9* (1/10)/(9/10) = 9*1/10*10/9 = 1
    my answer was marked correctly

    I don't see how my answer to the second part is wrong. I hope somebody can clear up this confusion for me. Thanks for any help anyone can provide me.
     
  2. jcsd
  3. Oct 28, 2011 #2

    Dick

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    Science Advisor
    Homework Helper

    I think your answer to the first part is also wrong. 9*(1/10)^n isn't a partial sum of the series. It's the nth term of the series.
     
  4. Oct 28, 2011 #3
    shouldn't the limit of the partial sums be the same as the sum of the series?
     
  5. Oct 28, 2011 #4
    I think your first part is okay. They didn't ask for the partial sum, they asked for the sequence of partial sums.

    [EDIT] never mind, it does seem wrong.
     
  6. Oct 28, 2011 #5

    Mark44

    Staff: Mentor

    There are two different sequences here. The first is {.9, .09, .009, ..., 9 * 10-n, ...}
    The sequence of partial sums is {.9, .99, .999, ...}
     
  7. Oct 28, 2011 #6
    Okay I found the sequence of partial sums to be [tex] S_n= \frac{9(1+10^{n-1})}{10^n} [/tex]

    Then I found the limit of that to be 9/10.

    Of course, this may easily be wrong. I'm just making an attempt.
     
  8. Oct 28, 2011 #7
    Actually, I think I MUST be wrong about 9/10, since isn't the limit of the sequence of partial sums supposed to equal to the sum of the series?

    Oh, I see a problem with my partial sums. Oh well. Your turn GreenPrint. :)
     
  9. Oct 29, 2011 #8

    Mark44

    Staff: Mentor

    Yes, let's let GreenPrint chime in now.
     
  10. Oct 29, 2011 #9

    HallsofIvy

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    Staff Emeritus
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    It might be much simpler to recognize that .9+ .09+ .009+ .0009+ ... is the same as .99999.... where the "9" continues for ever. What very simple number is that?
     
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