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Math Help - norm convergence law question

  1. #1
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    norm convergence law question

    i noticed in this problem

    that the norm gives us a series which is diverging.

    then they proove that its a couchy series so its converging

    why?????(i thought that if the sum of te norm is dievrgant then thats it,its divergant finily)
    what is the law behind this whole thing?

    Last edited by transgalactic; April 12th 2012 at 10:58 AM.
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  2. #2
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    Re: norm convergence law question

    We have \left\|\sum_{i=1}^n f_i\right\|\le\sum_{i=1}^n\|f_i\|, so even when the right-hand side diverges, the left-hand side may still converge. E.g., if f_n=\frac{(-1)^n}{n}\in L^2([0,1]), then \|f_n\|=1/n and \sum_{i=1}^\infty \|f_i\| diverges but \sum_{i=1}^\infty f_i converges by the alternating series test. This is similar how absolute convergence implies convergence but not vice versa for numerical series.

    The figure in your post (at least the formulas) do not finish the proof that \|S_m-S_n\| is small, i.e., that S_n is a Cauchy sequence.
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  3. #3
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    Re: norm convergence law question

    but its not smaller then diverging series
    its equal to a diverging series.

    so if its equal to a diverging series,then that it.

    how can they change that by proving that its a couchy series.

    (i know the law that if its a couchy series and its on L2 then its converging)
    but here it looks like they cheat because at first they showed that its diverging ,but then they change that around.

    this is the rest of the proof.

    Last edited by transgalactic; April 12th 2012 at 02:09 PM.
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