Question

Σ 1/(nlnn) from infi to n=2

Answer 1

it the question "does it converge"?

Answer 2

from n=2 to inf ?

Answer 3

because it seem unlikely

Answer 4

The question is does this convergent or divergent

Answer 5

try \[\int_2^{\infty}\frac{dx}{x\ln(x)}\]

Answer 6

Yes n=2 to infi

Answer 7

use the integral test

Answer 8

and the integral is done by a u - sub \(u=\ln(x)\)

Answer 9

Du = 1/x dx

Answer 10

yup, right in front of you, just what you need

Answer 11

So 1/u du

Answer 12

hmmm yup

Answer 13

Is this hormonic series?

Answer 14

it is an integral not a series

Answer 15

Oh

Answer 16

and no, harmonic series is \(\sum \frac{1}{n}\)
compute the anti - derivative and see what you get

Answer 17

\[\int \frac{du}{u}=?\]

Answer 18

Hm let me see

Answer 19

Ln u

Answer 20

k right
and \(u=\ln(x)\) so final answer?

Answer 21

Ln(lnx)?

Answer 22

yes

Answer 23

final job it so compute
\[\lim_{x\to \infty}\ln(\ln(x))\]

Answer 24

Then how am I define if it con or div

Answer 25

Oh do limit

Answer 26

Ln(ln(infi)) =infi

Answer 27

yes
so the integral diverges, and therefore so does the series

Answer 28

Ah I see so it div

Answer 29

Thank you so much sir