Is this series convergent or divergent

In summary, the debate is about whether the series Ʃ((-1)^(n-1)*ln(n))/n is convergent or divergent. The attempted solutions include using the p test, comparison tests, and the alternating series test. One person argues that the series is divergent due to the slow growth rate of ln(n) compared to n, while the other argues that the limit of ln(n)/n is zero and the series is convergent. The expert suggests using the alternating series test and clarifies the conditions for using comparison tests. The discussion ends with one person planning to research the topic further.
  • #1
Windowmaker
68
0

Homework Statement



Me and my friend are debating on wether the follow seris is convergent or divergent. The seris is the sum of (-1)^n-1 * ln(n)/n.

Homework Equations


p test and comparision tests.

And alternating series test

The Attempt at a Solution


My approach to this problem was that the ln(n) portion grows much slower than n portion. So I compared this function to 1/n. I know 1/n is divergent, so I concluded the above function was also divergent. My friend argues that the limit of ln (n)/n is zero and is greater than ln(n+1)/(n+1). So he says its convergent.
 
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  • #2
What about the (-1)^(n-1) factor? I think you'd better look at the alternating series test.
 
  • #3
Windowmaker said:

Homework Statement



Me and my friend are debating on wether the follow seris is convergent or divergent. The seris is the sum of (-1)^n-1 * ln(n)/n.

Homework Equations


p test and comparision tests.

And alternating series test

The Attempt at a Solution


My approach to this problem was that the ln(n) portion grows much slower than n portion. So I compared this function to 1/n. I know 1/n is divergent, so I concluded the above function was also divergent. My friend argues that the limit of ln (n)/n is zero and is greater than ln(n+1)/(n+1). So he says its convergent.
What does the alternating series test say in this case? You seem to be ignoring the fact that this is an alternating series.
 
  • #4
So by using the alternating series test, this seris is convergent?
 
  • #5
Also, disregarding for the moment that the series is alternating, when you compare ln(n)/n with 1/n, what must happen for you to conclude that Ʃ(ln(n)/n) diverges?
 
  • #6
They would have to grow at the same rate?
 
  • #7
Windowmaker said:
They would have to grow at the same rate?
No, that doesn't have anything to do with the comparison test, which is one of the tests that you listed, and are apparently attempting to use.

This test should be defined in your book. Specifically, there are different inequalities that come into play, depending on whether you are comparing to a convergent series or to a divergent series.
 
  • #8
Im confused. I am going to go youtube this. Have a nice day.
 

FAQ: Is this series convergent or divergent

What is the definition of convergence and divergence in a series?

Convergence and divergence refer to the behavior of a series, or a sum of terms, as the number of terms increases to infinity. A series is said to be convergent if the sum of its terms approaches a finite value as the number of terms increases. Conversely, a series is divergent if the sum of its terms does not approach a finite value as the number of terms increases.

How can I determine if a series is convergent or divergent?

There are several tests that can be used to determine the convergence or divergence of a series. These include the comparison test, ratio test, root test, and integral test. Each of these tests has specific criteria that must be met for a series to be convergent or divergent.

What is the purpose of determining convergence or divergence in a series?

Determining whether a series is convergent or divergent is important in understanding the behavior and properties of the series. It can also be used to evaluate the sum of an infinite series, which can have practical applications in fields such as physics, engineering, and economics.

Can a series be both convergent and divergent?

No, a series cannot be both convergent and divergent. A series can only have one of these two behaviors. However, it is possible for a series to be neither convergent nor divergent, meaning it does not have a defined behavior as the number of terms increases.

Are there any general rules for determining convergence or divergence in a series?

There are no general rules for determining convergence or divergence in a series. Each series is unique and must be evaluated using specific tests and criteria. It is important to carefully consider the properties and behavior of a series before determining its convergence or divergence.

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