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Limit Problem

  1. Apr 17, 2005 #1
    Can anyone help with these?

    1. [tex]\lim_{n\rightarrow 0}\frac{\log_{2}\sum_{k=1}^{2^n}\sqrt{k}}{n}[/tex]

    2. [tex]\lim_{n\rightarrow 0}\frac{\log_{2}\sum_{k=1}^{2^n}\sqrt{2k-1}}{n}[/tex]

    Thanks for you help.
  2. jcsd
  3. Apr 17, 2005 #2


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    Regarding just the first one:

    Although I question this, Mathematica returns:

    [tex]\mathop \lim\limits_{n\to 0}\frac{Ln[\sum_{k=1}^{2^n}\sqrt{k}]}{n}\approx 0.8527[/tex]

    Seems to me that it should be [itex]-\infty[/itex]

    Since once n gets below 1, the sum goes to just 1.

    How about this also too?

    [tex]\mathop \lim\limits_{n\to \infty}\frac{Ln[\sum_{k=1}^{2^n}\sqrt{k}]}{n}[/tex]

    This one Mathematica returns 1.0397 which makes sense if you plot the values for a range of n, it seems to approach a value near this.

    I'd like someone to explain these also.
  4. Apr 17, 2005 #3
    I encountered these problems on another site, and was just interested in them. I know some single-variable calculus, but I am just working on sequences and series now, so I was wondering if anyone knew of a trick to get these done.

    Thanks for your help.
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