# Is this series convergent or divergent

• Windowmaker
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.

## 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 about the (-1)^(n-1) factor? I think you'd better look at the alternating series test.

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.

So by using the alternating series test, this seris is convergent?

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?

They would have to grow at the same rate?

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.

Im confused. I am going to go youtube this. Have a nice day.