• Support PF! Buy your school textbooks, materials and every day products Here!

Does 1/n(log(n))^2 converge or diverge

  • #1

Homework Statement



Does 1/[n(log(n))^2] converge or diverge

Homework Equations



We know that Does 1/[n(log(n))] diverges by integral test


The Attempt at a Solution

 

Answers and Replies

  • #2
33,085
4,792
Then the comparison test might be useful, as long as you can show that 1/[n(log(n))^2] < 1/[n(log(n))].
 
  • #3
No sir, 1/[n(log(n))] diverges, comparison test would not help in this case.
 
  • #4
33,085
4,792
Sorry, misread what you wrote, which was clear. Does the ratio test help you? If not, try the limit comparison test.
 
  • #5
Most unfortunately both ratio test and limit comparison test give you 1 which is inconclusive.
 
  • #6
258
0
What's wrong with the integral test?
 
  • #7
Char. Limit
Gold Member
1,204
12
EDIT: Never mind, I think I misread the question.

Just to make sure, it's x log^2(x), not (x log(x))^2, right?
 
  • #8
258
0
Have you tried integrating [tex]\int \frac{dx}{x^2 log^2(x)}[/tex]?

It requires the Exponential Integral function Ei(u) to even be possible.
From the way he wrote it, it looks like it should be

[tex]
\frac{1}{n (log(n))^2}
[/tex]

But if you are right that he meant x^2 log^2(x) then it is trivial.

Hint: log^2(x) > 1 for x > 3.
 
  • #9
Unfortunately I have never tried the Ei(u) thing. Nor have I heard of integral function.
In other words 1/[n(log(n))^2] diverges?
 
  • #10
Yes, the integral test, however please tell me how to evaluate this integral? I suspect the result would be some expression that goes to infinity.
 
  • #11
Char. Limit
Gold Member
1,204
12
Wait, before we continue.

We need to know your bounds.

Is the initial n n=1?

Or is the initial n n=2?
 
  • #12
258
0
How to evaluate
[tex]
\int_{2}^{\infty} \frac{1}{x log^2 (x)} dx
[/tex]

that?

Think back to Calc II. Try a u-substitution.
 
  • #13
352
0
You can also use a "bare-hands" argument, without the integral test. Imagine that the natural logarithm were instead a binary logarithm (it's just a constant factor different), and estimate the sequence by blocks whose boundaries are powers of 2.
 
  • #14
initial n=2
 
  • #15
Char. Limit
Gold Member
1,204
12
initial n=2
Then an integral test will work just fine. Just do as l'Hopital suggested.
 
  • #16
and therefore it diverges?
 
  • #17
2,967
5
Use integral test and method of substitution.
 

Related Threads for: Does 1/n(log(n))^2 converge or diverge

Replies
4
Views
21K
Replies
3
Views
3K
Replies
2
Views
1K
  • Last Post
Replies
7
Views
3K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
5
Views
41K
  • Last Post
Replies
2
Views
519
  • Last Post
Replies
9
Views
8K
Top