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

## 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

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Mark44
Mentor
Then the comparison test might be useful, as long as you can show that 1/[n(log(n))^2] < 1/[n(log(n))].

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

Mark44
Mentor
Sorry, misread what you wrote, which was clear. Does the ratio test help you? If not, try the limit comparison test.

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

What's wrong with the integral test?

Char. Limit
Gold Member
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?

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.

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?

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
Gold Member
Wait, before we continue.

We need to know your bounds.

Is the initial n n=1?

Or is the initial n n=2?

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

that?

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

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.

initial n=2

Char. Limit
Gold Member
initial n=2
Then an integral test will work just fine. Just do as l'Hopital suggested.

and therefore it diverges?

Use integral test and method of substitution.