Question

absolute converges or conditional converges?
n=3, ln(ln n) / n (ln n)

Answer 1

\[\sum_{3}^{\infty} (-1) ^{n+1} \frac{ \ln(\ln n) }{ n(\ln n) }\]

Answer 2

To help you get started:
First, test for absolute convergence. Ignore the (-1)^(n+1) factor and concentrate on

ln(ln n)
_______ .
n(ln n)

We'll need to determine whether three conditions are satisfied: (a) Is this expression positive for n > N? [Yes, it is, beginning with n=3.] (b) Is this expression always decreasing for n > N? [Yes, it is, beginning with n = 6.] (c) If we take the limit as n approaches infinity, does this expression approach zero? [Yes, it does.]

Therefore this sequence is absolutely convergent.

Answer 3

You can determine whether a function is increasing or decreasing by taking its derivative. For values of the independent variable for which the derivative is negative, the function is decreasing. That's the sophisticated way of doing this. But you could also use a graphing calculator to observe the behavior of the function as its independent variable increases.

Answer 4

it is not absolutely convergent..though it is conditionally convergent

Answer 5

Care to explain how you arrived at these conclusions?

Answer 6

use the comparison test/integral test...then the alternating series test