Does (-1)^(n)ln(n)/n Converge Conditionally?

[frasifrasi]

is

ln(n+1)/(n+1) smaller than ln(n)/(n)

I am trying to show that (-1)^(n)ln(n)/n converges conditionally...
Thank you.

P.s the summation is from 1 to infinity.

 

[Dick]

Is log(x)/x a decreasing function?

 

[dynamicsolo]

is

ln(n+1)/(n+1) smaller than ln(n)/(n)

I am trying to show that (-1)^(n)ln(n)/n converges conditionally...
Thank you.

P.s the summation is from 1 to infinity.

You might find it easier, since you're looking at the Alternating Series Test, to just check the limit as n--> infinity of ln(n)/n . L'Hopital's Rule tells you at once that it's zero.

I've done this one myself and I'm not sure there's a nice way to see the inequality you noted. (And the Ratio Test is unhelpful once again...)

 

[frasifrasi]

SO how would I do it?

I have established that the abs value diverges... so, I have to use something else to prove it's conditionally (this series is supposed to converge conditionally). How can I do this?

 

[Dick]

Prove ln(x)/x is a decreasing function for large x. Take the derivative and see if it's negative.

 

[frasifrasi]

dick, that sounds right.

I got (1 - ln(x))/x^(2)

-- how do I go about showing that this is negative?

 

[frasifrasi]

It would be great if you could show me how to get there. I know I am supposed to figure out on my own by my test is tomorrow and I am trying to learn as much as I can. I want to make sure i know this series before the test.

 

[frasifrasi]

I suppose that the ln(x) will outgrow that 1, so the numerator will eventually become negative, so the series is decreasing and the alternating test says it converges. Is this right?

-- Also, can you take a look at the Urgent post I just made ? : )

 

[Dick]

Figure out when it's zero or undefined. Those are the only places it can change sign. Then test the intervals in between. This is the same old problem of trying to figure intervals where a function is increasing and decreasing. You must have done some of this. You don't even need to be that detailed. What can you say about the derivative if x>e?

 

[Dick]

I suppose that the ln(x) will outgrow that 1, so the numerator will eventually become negative, so the series is decreasing and the alternating test says it converges. Is this right?

-- Also, can you take a look at the Urgent post I just made ? : )

I think I already did. Unless you posted something even more urgent.

 

[Dick]

I suppose that the ln(x) will outgrow that 1, so the numerator will eventually become negative, so the series is decreasing and the alternating test says it converges. Is this right?

-- Also, can you take a look at the Urgent post I just made ? : )

Yes, that's right.