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I Proof of series`s tail limit

  1. Mar 16, 2017 #1
    {\displaystyle \sum_{n=1}^{\infty}a_{n}}
    is converage, For N\in
    \mathbb{N}


    \sum_{n=N+1}^{\infty}an
    is also converage

    proof that \lim_{N\rightarrow\infty}(\sum_{n=N+1}^{\infty}an)=0
    Code (Text):
    {\displaystyle \sum_{n=1}^{\infty}a_{n}}
      is converage, For N\in
    \mathbb{N}

    \sum_{n=N+1}^{\infty}an
      is also converage

    proof that \lim_{N\rightarrow\infty}(\sum_{n=N+1}^{\infty}an)=0
     
     
    Last edited: Mar 16, 2017
  2. jcsd
  3. Mar 16, 2017 #2
    Can you write Σ1an - ΣN+1an as a sum?

    It might be helpful to put Σ1an = c .
     
  4. Mar 19, 2017 #3

    Mark44

    Staff: Mentor

    Please take a look at our LaTeX tutorial -- https://www.physicsforums.com/help/latexhelp/
    I believe this is what you were trying to convey:
    ##\sum_{n=1}^{\infty}a_{n}## converges.

    For ##N \in \mathbb{N}, \sum_{n = N+1}^{\infty}a_n## also converges.

    What does it mean for a series to converge? How is this defined?

    Also, is this a homework problem?
     
  5. Mar 19, 2017 #4

    mathman

    User Avatar
    Science Advisor
    Gold Member

    The difference between a full series and a tail is finite, so convergence is te same for both.
     
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