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

[grossgermany]

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

 

[Mark44]

Then the comparison test might be useful, as long as you can show that 1/[n(log(n))^2] < 1/[n(log(n))].

 

[grossgermany]

No sir, 1/[n(log(n))] diverges, comparison test would not help in this case.

 

[Mark44]

Sorry, misread what you wrote, which was clear. Does the ratio test help you? If not, try the limit comparison test.

 

[grossgermany]

Most unfortunately both ratio test and limit comparison test give you 1 which is inconclusive.

 

[l'Hôpital]

What's wrong with the integral test?

 

[Char. Limit]

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?

 

[l'Hôpital]

Have you tried integrating $$\int \frac{dx}{x^2 log^2(x)}$$?

It requires the Exponential Integral function Ei(u) to even be possible.

From the way he wrote it, it looks like it should be

$$\frac{1}{n (log(n))^2}$$

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.

 

[grossgermany]

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?

 

[grossgermany]

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.

 

[Char. Limit]

Wait, before we continue.

We need to know your bounds.

Is the initial n n=1?

Or is the initial n n=2?

 

[l'Hôpital]

How to evaluate
$$\int_{2}^{\infty} \frac{1}{x log^2 (x)} dx$$

that?

Think back to Calc II. Try a u-substitution.

 

[ystael]

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.

 

[grossgermany]

initial n=2

 

[Char. Limit]

initial n=2

Then an integral test will work just fine. Just do as l'Hopital suggested.

 

[grossgermany]

and therefore it diverges?

 

[Dickfore]

Use integral test and method of substitution.