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Math Help - Help with proving properties of Tails of Series

  1. #1
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    Help with proving properties of Tails of Series

    Hi, I have an issues with solving this problem.

    Let the summation of an from n=1 to infinity be a series and let m be in the natural numbers. Show that the summation of an from n=1 to infinity converges IFF the summation of am+n from n=1 to infinity converges, and that (the summation of an from n=1 to infinity) = Sm + (the summation of am+n from n=1 to infinity converges).

    I understand that this problem is basically telling you to prove the properties of the Tails of series, but I'm not sure how I would go about proving it. And sorry for all the long typing for all the summation things, I don't know how to type it out the proper way.
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Help with proving properties of Tails of Series

    Well, \displaystyle\sum_{n=1}^{\infty} a_n = \displaystyle\sum_{n=1}^{m} a_n +\displaystyle\sum_{n=1}^{\infty} a_{n+m}

    Now if \displaystyle\sum_{n=1}^{\infty} a_n = L , Then  \displaystyle\sum_{n=1}^{m} a_n +\displaystyle\sum_{n=1}^{\infty} a_{n+m}  = L .
    Since  \displaystyle\sum_{n=1}^{\m} a_n  =  k where k is a finite number(you are just adding up a finite number of terms so you get a finite number).
    \displaystyle\sum_{n=1}^{\infty} a_{n+m} = L - K (a finite term) . So \displaystyle\sum_{n=1}^{\infty} a_{n+m} converges.

    It should be pretty straightforward.
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  3. #3
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    Re: Help with proving properties of Tails of Series

    Hi thanks for responding. However, it says also to prove (the summation of an from n=1 to infinity) = Sm + (the summation of am+n from n=1 to infinity converges), which is basically what you wrote for the first line, so I feel like the answer you provided is incomplete. And sorry if this is a misguided response, but I'm still very lost in the whole field of real analysis still. so sorry and thank you.
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  4. #4
    Senior Member jakncoke's Avatar
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    Re: Help with proving properties of Tails of Series

    \displaystyle\sum_{n=1}^{m} a_n = a_1 + a_2 + ... + a_m
    \displaystyle\sum_{n=1}^{\infty} a_{n+m} = a_{m+1} + a_{m+2} + ...

    Now i don't know how to be more clear than that, surely adding those two u get  a_1 + a_2 + .... + a_m + a_{m+1} + .... . = \displaystyle\sum_{n=1}^{\infty} a_n
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  5. #5
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    Re: Help with proving properties of Tails of Series

    Okay, Thank you. I get it now .
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