Convergence of a Challenging Series with an Additional Factor

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In summary, the conversation is discussing how to solve the problem \sum from n=2 to \infty of n/((n2-5)*(ln n)2). Several methods have been attempted, including limit comparison and ratio test, but the integral test is suggested as a potential solution. Further explanation is requested and it is clarified that the problem also includes the factor 1/(ln n)^2. The conversation ends with a reminder to check that the integral exists and a hint for finding an explicit antiderivative.
  • #1
simba924
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Homework Statement



[tex]\sum[/tex] from n=2 to [tex]\infty[/tex] of n/((n2-5)*(ln n)2)

Homework Equations





The Attempt at a Solution


I've tried Limit Comparison but I always get a limit of 0 which will not work. Ratio test doesn't help. I don't think a direct comparison can be made but that seems to be the only other option...
 
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  • #2
Hi simba924!

To simplify the problem a bit, show that

[tex]\frac{n}{n^2-5}\le\frac{2}{n}[/tex]

for large n. Then apply the http://en.wikipedia.org/wiki/Integral_test_for_convergence" [Broken].
 
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  • #3
Ahh thanks. Can you explain that in a little more detail though, I don't really get it.
 
  • #4
Hey I still need some help with this one. I'm pretty sure that [tex]\sum[/tex] 2/n is not convergent because it is in form integer/np where p=1

Can anyone help?
 
  • #5
simba924 said:
Hey I still need some help with this one. I'm pretty sure that [tex]\sum[/tex] 2/n is not convergent because it is in form integer/np where p=1

Can anyone help?

Remember, you still have the additional factor 1/(ln n)^2 (without which it would diverge). To do the integral test you have to check that

[tex]\int_2^{\infty}\frac{1}{x(\log(x))^2}[/tex]

exists (you can find an explicit antiderivative, Hint: substitution).
 

What is "proving convergence"?

Proving convergence is a method used in mathematics and science to determine whether a sequence, series, or function approaches a specific value or behavior as its input or index increases. It involves analyzing the behavior of the sequence or series and providing evidence or a mathematical proof to support the claimed convergence.

Why is proving convergence important?

Proving convergence is important because it allows us to determine the behavior and limit of a sequence, series, or function, which can help us make predictions and draw conclusions about real-world phenomena. It also ensures the accuracy and validity of mathematical and scientific models and theories.

What are some common techniques used to prove convergence?

Some common techniques used to prove convergence include the comparison test, the ratio test, the root test, the integral test, and the limit comparison test. These techniques involve comparing the given sequence or series to a known one or using properties of limits to determine the convergence or divergence of the given sequence or series.

What are some challenges scientists face when proving convergence?

Some challenges scientists face when proving convergence include identifying the correct method or test to use, dealing with complex or infinite series, and ensuring the accuracy and validity of the proof. Additionally, certain functions or sequences may have unique behaviors that require more advanced techniques or approaches to prove convergence.

What are some real-world applications of proving convergence?

Proving convergence has various real-world applications, such as in engineering, physics, economics, and computer science. For example, engineers may use convergence to analyze the behavior of a system over time, physicists may use it to predict the behavior of a particle or wave, economists may use it to make predictions about market trends, and computer scientists may use it to analyze the efficiency of algorithms.

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