For what values of p does this series converge?

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In summary, the conversation discusses how to find the values of p for which the given series converges absolutely. Different tests, such as the ratio test, root test, and limit comparison test, have been tried but none have been successful in proving that the series converges. The suggestion to use a comparison test with 1/k is given, but it is ultimately determined that the series is not a p-series and therefore the p-test cannot be used.
  • #1
applegatecz
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Homework Statement


Find all values of p for which the given series converges absolutely: [tex]\sum[/tex] from k=2 to infinity of [1/((logk)^p)].


Homework Equations




The Attempt at a Solution


I've tried the ratio test, the root test, limit comparison test ... everything. I know the answer is the null set (that is, for no values of p does the series converge), but I can't prove that rigorously.
 
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  • #2
Do a comparison test with 1/k. Can you show lim k->infinity (log(k))^p/k=0?
 
  • #3
have you considered simply looking at this question as a "p-test"
 
  • #4
oinkbanana said:
have you considered simply looking at this question as a "p-test"
The series is not a p-series, so this test is not applicable. Here is a p-series:
[tex]\sum_{n = 1}^{\infty} \frac{1}{n^p}[/tex]
 

Related to For what values of p does this series converge?

1. What is the definition of convergence in a series?

Convergence in a series refers to the behavior of the terms in a series as the number of terms increases. A series is said to converge if the sum of its terms approaches a finite value as the number of terms increases, and diverges if the sum of its terms becomes infinitely large.

2. How do you determine if a series converges or diverges?

There are several tests that can be used to determine if a series converges or diverges, such as the ratio test, the root test, and the integral test. These tests involve evaluating the behavior of the terms in the series and comparing them to known convergence or divergence criteria.

3. What is the significance of the value of p in determining convergence?

The value of p is important because it determines the behavior of the terms in the series. For example, if p is less than 1, the series is likely to converge, while if p is greater than or equal to 1, the series is likely to diverge. However, this is not always the case and other tests must be used to determine convergence or divergence.

4. Can a series converge for some values of p and diverge for others?

Yes, a series can have different convergence behavior for different values of p. For example, the series ∑ n^p may converge for p = 2 but diverge for p = 3. This is because as p changes, the behavior of the terms in the series also changes, resulting in different convergence or divergence behavior.

5. How do you use the ratio test to determine convergence for a series?

The ratio test involves taking the limit of the ratio of the absolute value of the (n+1)th term to the nth term as n approaches infinity. If this limit is less than 1, the series converges. If it is greater than 1, the series diverges. If the limit is exactly 1, the test is inconclusive and other tests must be used to determine convergence or divergence.

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