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Math Help - [SOLVED] sums exist or do not exist

  1. #1
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    [SOLVED] sums exist or do not exist

    Hi,can some one please help me with the following sums as i need to prove that they either exist or do not exist:

    We do NOT say the sum exists if it is infinity.

    1.  \sum _{k=1}^{\infty }{{k}^{-1}-{k}^{-2}}

    I know that:

    this sequnce \sum _{k=1}^{\infty }{k}^{-1} doesnot exist

    but this sequence \sum _{k=1}^{\infty }-{k}^{-2}does exist

    so i am confused about if this sequence \sum _{k=1}^{\infty }{{k}^{-1}-{k}^{-2}} exists

    2. i am not sure if the following sum exists


    \sum _{k=1}^{\infty }{{k}^{-1}- \left( k+1 \right) ^{-1}}
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  2. #2
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    The first one does not exist since if it did \sum_{k=1}^{\infty } (k^{-1} + k^{-2}) -\sum_{k=1}^{\infty } k^{-2} = \sum_{k=1}^{ \infty } k^{-1} would be finite.

    For the second one notice that \sum_{k=1}^{n} k^{-1} -(k+1)^{-1} = (1 -1/2)+(1/2-1/3)+...+(1/n-1/(n+1))=1-1/(n+1)
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  3. #3
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    Do you know the limit comparison test?
    Compare \frac{1}{n}-\frac{1}{n^2} with \frac{1}{n}.
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  4. #4
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    Quote Originally Posted by Plato View Post
    Do you know the limit comparison test?
    Compare \frac{1}{n}-\frac{1}{n^2} with \frac{1}{n}.
    we are not able to use the the limit comparison test, since my lecture has not proved it.

    However we are able to use the comparison test and the absolute convergence, but i don't know if that will help.
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  5. #5
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    THANKS FOR THE HELP Plato AND Jose27
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