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Math Help - Trying to clarify a detail in Sum Of Series Theory..

  1. #1
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    Trying to clarify a detail in Sum Of Series Theory..

    Hello everyone,

    I have checked some post about sum here but...

    There is this Serie:

    (2n+1)/n^2(n+1)^2
    with n=1 to infinity

    Actually this is an easy one, I do partial fraction decomposition and come up with 1 - 1/(n+1)^2 and the limit of the last is the sum of the Serie (I am now learning series).

    However I have a question, why I cannot go directly take the limit from the very begining, which results to zero and say that the sum is zero.

    I know this is not right, but I cannot really express it why, taking the sum even for n=2 it immediately shows that my guess is wrong. Is there something more to give as an answear to my question?

    Thank you all guys,
    and I am sorry if it is not a very clear question!
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Trying to clarify a detail in Sum Of Series Theory..

    Quote Originally Posted by Melsi View Post
    Hello everyone,

    I have checked some post about sum here but...

    There is this Serie:

    (2n+1)/n^2(n+1)^2
    with n=1 to infinity

    Actually this is an easy one, I do partial fraction decomposition and come up with 1 - 1/(n+1)^2 and the limit of the last is the sum of the Serie (I am now learning series).

    However I have a question, why I cannot go directly take the limit from the very begining, which results to zero and say that the sum is zero.

    I know this is not right, but I cannot really express it why, taking the sum even for n=2 it immediately shows that my guess is wrong. Is there something more to give as an answear to my question?

    Thank you all guys,
    and I am sorry if it is not a very clear question!
    Look at:  \sum_{n=1}^{\infty}\frac{1}{n}


    Moreover:

    Necessarily condition for series convergence:

    If you have a converges series:  \sum_{n=1}^{\infty}a_n then :

     \lim_{n\to\infty}a_n=0


    The opposite direction will not work see the above.


    By the way... \frac{2n+1}{n^2(n+1)^2} \neq 1 - \frac{1}{(n+1)^2} .


    \frac{2n+1}{n^2(n+1)^2}=\frac{1}{n^2}-\frac{1}{(n+1)^2}

    And now read here: Telescoping series - Wikipedia, the free encyclopedia
    Last edited by Also sprach Zarathustra; June 16th 2011 at 04:26 PM.
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  3. #3
    Junior Member
    Joined
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    Athens, Greece
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    Re: Trying to clarify a detail in Sum Of Series Theory..

    I like this part: '... always comes of ignorance'.

    Yes this is a very good point 1/n does converge but its serie does not, very good connection!

    This shows that the limit of the general term (1/n) is a different thing on its own from the sum of the serie. This is the point where I messed things up (still sequences running in my head). The limit of the general term is an other thing from the sum of the serie it self and they are not equal. Thank you very very much for your answer it is so clear now!!!
    Last edited by Melsi; June 17th 2011 at 10:35 AM.
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