Limit Problem

  • Thread starter amcavoy
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  • #1
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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.
 

Answers and Replies

  • #2
saltydog
Science Advisor
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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.
 
  • #3
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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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