Is the series 1/(nln) from n=2 to infinity convergent or divergent? Use the integral test.

Question

I got stuck when I took the derivative. What should I do to satisfy that it is continuousmy positive, and decreasing? Thanks.

Answer 1

Do you know how your teachers and the textbooks says things are obvious...they don't really want you to prove it's decreasing and positive, but rather look, by inspection.

 

So I'm going to look at this as 1/n * 1/ ln(n).   Since n is always increasing, and so is ln n, then 1/n and 1/ln n is always decreasing. and since n is positive, they are both always positive...ln n is always positive fir values greater than 1.

 

So it satisfies the conditions for the integral test.

 

So the integral of that needs u=ln n so du = 1/n dn   resulting in  ∫1/u du = ln ⌈u⌉ = ln (ln x)  which from 2 to infinity, diverges.

 

Hope this helped.