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Limit of a Series

  1. Oct 31, 2006 #1
    I have the following problem:
    [​IMG]
    I know for a fact that the answer is not 0, but I have no idea how to actually find the answer. I've never seen a similar problem before, and I'm not really sure how to start it.
     
  2. jcsd
  3. Oct 31, 2006 #2

    StatusX

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    Can you find a function such that if you approximate its integral over some range by n rectangular strips, you get that sum? Then the limit would just be the integral over that range.
     
  4. Oct 31, 2006 #3

    quasar987

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    I think there's a more direct approach.

    Remember that a series is a sequence of partial sums. The general term of the sequence is

    [tex]a_n=\sum_{k=1}^{n}\frac{k}{n^2} = \frac{1}{n^2}\sum_{k=1}^{n} k=?[/tex]

    You know how to do that sum on the left, and thus you can find an explicit form of the general term. Just take the limit.
     
  5. Oct 31, 2006 #4
    I'm not really sure what you mean by the explicit form of the general term. Can you give me a little more help, please?
     
  6. Oct 31, 2006 #5

    quasar987

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    Do you know what

    [tex]\sum_{k=1}^{n} k[/tex]

    sums to? It's sometimes called "Gauss' sum" named after the 8 years old who found the value of the sum in his head when asked to compute the sum of the first 50 integers. :smile:
     
  7. Oct 31, 2006 #6

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    So you can get that sum directly in a few different ways, the simplest of which is to consider the pictures:

    *0

    **0
    *00

    ***0
    **00
    *000

    etc.

    If you know that formula, that's definitely the easiest way. What I was suggesting might be overkill, and I was thinking of it because it's the way you would go about this if you had a higher power of k, ie:

    [tex]\sum_{k=1}^{n}\frac{k^p}{n^{p+1}} [/tex]
     
  8. Oct 31, 2006 #7
    Oh, duh! I can't believe I didn't think of that. Thanks!
     
  9. Oct 31, 2006 #8

    quasar987

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    Generally, given an infinite series

    [tex]\sum_{k=1}^{+\infty}b_n[/tex]

    you cannot put the general term [itex]a_n=\sum_{k=1}^n b_n[/itex] of the sequence of partial sums in a friendly form that allows for direct computation of the value of the series by just taking the limit:

    [tex]\lim_{n\rightarrow +\infty}a_n[/tex].

    But the series you're dealing with is one of these rare case where the general term has a friendly form in terms of n that allows for this method of calculating the sum to work.
     
    Last edited: Oct 31, 2006
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