# Is this series convergent or divergent

## 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.

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

Mark44
Mentor

## 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?

Mark44
Mentor
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?

Mark44
Mentor
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. Im gonna go youtube this. Have a nice day.