Question

Determine whether the series is convergent or divergent \[\sum_{2}^{\infty} 1\div \left( n \left( \ln n \right)^{2} \right)\]

Answer 1

It's convergent. By using the integral test:\[\int \frac{1}{x (\log{x})^2}dx = -\frac{1}{\log{x}}+C\]so\[\int_2^\infty \frac{1}{x (\log{x})^2}dx = \frac{1}{\log{2}} < \infty.\]