Analyzing Convergence of 1/n(ln n)^p and 1/((n(ln n)(ln (ln n)))^p

In summary, the question is for what value of p>0 does the series 1/(n(ln n ) ^p) and 1/((n(ln n)(ln (ln n))))^p converges and for what value does it diverges, with a=1 and b=infinity for the integral signs. The integral test and ratio test can be used to determine the convergence or divergence of the series, with the criterion \sum a_n converges iff \sum 2^n a_{2^n} converges being particularly useful for series involving ln(n). It is also important to note that the series should be monotonically decreasing for this method to work.
  • #1
jkh4
51
0
Hi, I just wondering if someone can help me on this question.

1) 1/(n(ln n ) ^p)

2) 1/((n(ln n)(ln (ln n))))^p

Both are b = infinity and a = 1 for the integral signs.

The question is for what value of p>0 does the series converges and for what value does it diverges.

Thank you !
 
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  • #2
There was a similar question in the HW section recently. Use the integral test.
 
  • #3
Do you know about the criterion [itex]\sum a_n[/itex] converges iff [itex] \sum 2^n a_{2^n}[/itex] converges? It is usefull for series involving ln(n) because ln(2^n)=nln(2) !

Then, for the values of p, use the ratio test (a_{n+1}/a_n).
 
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  • #4
quasar987 said:
Do you know about the criterion [itex]\sum a_n[/itex] converges iff [itex] \sum 2^n a_{2^n}[/itex] converges?

It's important to keep in mind this only works when the series is monotonically decreasing (which this one is). Otherwise, just take the terms to be all 1's except for 0's when n is a power of 2.
 

Related to Analyzing Convergence of 1/n(ln n)^p and 1/((n(ln n)(ln (ln n)))^p

Question 1: What is the purpose of analyzing the convergence of 1/n(ln n)^p and 1/((n(ln n)(ln (ln n)))^p)?

The purpose of analyzing the convergence of these two sequences is to understand their behavior as n approaches infinity. This information can be useful in various fields of science, such as statistics, physics, and computer science.

Question 2: How do you determine the convergence of a sequence?

To determine the convergence of a sequence, we look at its limit as n approaches infinity. If the limit exists and is a finite number, the sequence is said to be convergent. If the limit does not exist or is infinite, the sequence is said to be divergent.

Question 3: What is the relationship between the values of p and the convergence of 1/n(ln n)^p and 1/((n(ln n)(ln (ln n)))^p)?

The values of p play an important role in determining the convergence of these two sequences. If p is greater than 1, both sequences will converge. If p is equal to or less than 1, both sequences will diverge. Additionally, as p increases, the rate of convergence also increases.

Question 4: How do you analyze the convergence of 1/n(ln n)^p and 1/((n(ln n)(ln (ln n)))^p)?

To analyze the convergence of these sequences, we can use various mathematical techniques such as the ratio test, comparison test, or the integral test. These tests help us determine the behavior of the sequence as n approaches infinity.

Question 5: What are the practical applications of understanding the convergence of 1/n(ln n)^p and 1/((n(ln n)(ln (ln n)))^p)?

Understanding the convergence of these sequences can have practical applications in fields such as data analysis, optimization problems, and computational algorithms. It can also help in predicting the behavior of a system or process as it approaches a large number of iterations.

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