TITLE: Finding if $\sum_{}^{}\frac{\ln k}{k}$ converges or diverge
QUESTION [6 upvotes]: How can I find whether the serie
$$\sum_{}^{}\frac{\ln k}{k}$$
converge or diverge using the basic comparison test or limit test.
The part that is confusing me is how to deal with $\ln$.

REPLY [1 votes]: Another way would be to use the integral test. The series
$$
\sum_{k = 1}^{\infty} \frac{\ln(k)}{k}
$$
converges if and only if the integral
$$
\int_1^{\infty} \frac{\ln(x)}{x} \,dx
$$
converges. Letting $u = \ln(x)$ we get $du = \frac{1}{x} \,dx$ so we obtain
$$
\int_{0}^{\infty} u \, du = \left[ \frac{u^2}{2} \right]^{\infty}_{0} = \lim_{t \to \infty} \left[ \frac{u^2}{2} \right]^t_0
$$
which tends to infinity, implying that the series is divergent.
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