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Simple series proof

  1. Nov 6, 2013 #1
    show that if ## a_n ## is any sequence and m is any integer such that the series ## \displaystyle \sum_{n=m}^\infty a_n ## converges, then ## \displaystyle\lim_{k \to \infty} \sum_{n=k}^{\infty
    } a_n = 0 ##

    let ## S_N = \displaystyle \sum_{n=m}^N a_n ##

    ## \lim_{N \to \infty} S_N = l ## from definition
    also,
    ## \lim_{N \to \infty} S_{N-1} = l ##

    ## S_N - S_{N-1} = a_N ##

    ## \lim_{N \to \infty} (S_N - S_{N-1}) = \lim_{N \to \infty}(S_N) - \lim_{N \to \infty} S_{N-1} = 0 ## then ## \lim_{N \to \infty} a_N = 0 ## i.e. ## \lim_{N \to \infty} \sum_{n=m}^N a_n = 0 \Rightarrow \lim_{k \to \infty} \sum_{n=k}^{\infty} a_n = 0 ##

    as you can see I don't really know how to prove this statement, but I have attempted it.

    Could someone show me how I can proceed to do the proof properly?
     
  2. jcsd
  3. Nov 6, 2013 #2

    pasmith

    User Avatar
    Homework Helper

    You are given that
    [tex]
    \sum_{n=m}^\infty a_n
    [/tex]
    converges. Thus you have
    [tex]
    L = \sum_{n=m}^\infty a_n = S_{k-1} + \sum_{n = k}^{\infty} a_n
    [/tex]
    so that
    [tex]
    |L - S_{k-1}| = \left| \sum_{n = k}^{\infty} a_n \right|
    [/tex]
     
  4. Nov 7, 2013 #3
    how did you get ## L = \sum_{n=m}^\infty a_n = S_{k-1} + \sum_{n=k}^{\infty} a_n ##?
     
  5. Nov 7, 2013 #4

    FeDeX_LaTeX

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    Gold Member

    They've just written it as a sum of two series.
     
  6. Nov 7, 2013 #5
    I do not believe it is correct though.
    It should read [itex]L=\sum_{n=0}^\infty a_n = S_{k-1}+\sum_{n=k}^\infty a_n[/itex].
    This is to be seen from the definition of [itex]S_k = \sum_{n=0}^k a_n[/itex].

    The final step in post #2 is correct. (Although I'm not entirely sure why the absolute value shows up, but it doesn't make any difference here)
     
  7. Nov 7, 2013 #6

    pasmith

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    Homework Helper

    Read the OP. We are told that [itex]\sum_{n=m}^\infty a_n[/itex] converges for some integer [itex]m[/itex]. In particular, we are not told that [itex]m[/itex] is positive. Therefore the OP's definition of [itex]S_k = \sum_{n=m}^{k} a_n[/itex] is correct.

    However, in any event the behaviour of a finite number of terms at the beginning of a sequence does not affect convergence of the series.
     
  8. Nov 7, 2013 #7
    My bad, sorry for the confusion.
    But this doesn't change anything indeed.
     
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