It is my understanding that for the integral test to apply, the series a must be decreasing, positive, and continuous over [n, ∞) but one of the problems I've encountered seems to break the first rule, yet my textbook still says it converges. I don't know what I am doing incorrectly.

Summation of [ln(n)]/(n^{2}) starting at n=1, counting to infinity

f(x) = [ln(n)]/(n^{2})

f'(x) = [1 - 2ln(n)]/(n^{3})

0 = 1 - 2ln(n)

n = e^{.5 }= 1.645

When you do a sign test on either side, you discover that after, to the right, f'(x) is negative, which to the left it is positive. This would leave me to believe that you could not apply the integral test here, yet the book I'm using put this in with the section about the integral test and says it diverges, which leads me to believe I'm missing something.

Thanks

PS sorry for the lack of formatting, I tried to get it to work, but it wasn't cooperating.