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

1 Answer
Jim H
Aug 30, 2015

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.

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