Solving Limit Problems: Help Needed

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In summary, the conversation revolves around two limits: \lim_{n\rightarrow 0}\frac{\log_{2}\sum_{k=1}^{2^n}\sqrt{k}}{n} and \lim_{n\rightarrow 0}\frac{\log_{2}\sum_{k=1}^{2^n}\sqrt{2k-1}}{n}. The person questioning these limits believes that the first one should be -\infty, but Mathematica returns 0.8527. The second limit approaches 1.0397 as n approaches infinity. The person is looking for an explanation or trick to solve these limits.
  • #1
amcavoy
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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.
 
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  • #2
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
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.
 

1. What are limit problems?

Limit problems involve finding the value that a function approaches as the input approaches a specific value. This is commonly used in calculus to analyze the behavior of a function near a certain point.

2. How do I solve limit problems?

There are various techniques for solving limit problems, including direct substitution, factoring, and using trigonometric identities. It is important to also consider the properties of limits, such as the squeeze theorem and the limit laws.

3. What are some common mistakes when solving limit problems?

Some common mistakes include incorrectly applying limit laws, forgetting to check for removable discontinuities, and not considering the behavior of the function as it approaches the specific value.

4. Can limit problems be solved using technology?

Yes, limit problems can be solved using technology such as graphing calculators or online limit calculators. However, it is important to understand the concepts and techniques behind solving limit problems rather than relying solely on technology.

5. How can I check if my answer to a limit problem is correct?

You can check your answer by graphing the function and observing the behavior near the specific value. You can also use technology to verify your answer or consult with a math teacher or tutor for confirmation.

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