Question

Using the integral test, how do you show whether `sum 3/(n sqrt(ln(n)))` diverges or converges from n=1 to infinity?

Answer 1

See the explanation.

Explanation:

Let `f(x) = 3/(xsqrtln(x))`. `" "`(So that `f(n) = a_n`.)

It is clear the `f` is continuous and decreasing on `[1,oo)`

so

`sum 3/(n sqrt(ln(n)))` converges or diverges

along with `int_1^oo 3/(xsqrtln(x)) dx`

`int_1^oo 3/(xsqrtln(x)) dx = lim_(brarroo)int_1^b 3/(xsqrtln(x)) dx`

`= 3lim_(brarroo)int_1^b (ln(x))^(-1/2) 1/x dx`

`= 3lim_(brarroo) 2(ln(x))^(1/2)]_1^b`

`= 6lim_(brarroo) (sqrtln(b) - sqrtln(1))`

`= 6lim_(brarroo) sqrtln(b) = oo`

The integral and the series diverge.