The series is alternating, suggesting the Leibniz test. For this, $\left\{\ln \left(n\right)\/n\right\}$ must eventually become a monotone decreasing sequence with limit zero. Consequently, the following calculations are executed.

$\underset{n\to \infty }{lim}\ln \left(n\right)\&sol;n$ = $0$  and   $\frac{d}{\mathrm{dx}}\left(\ln \left(x\right)\&sol;x\right)\&equals;\frac{1- \ln \left(x\right)}{x^2}<0$ for $x\&gt;e\doteq 2.7$

The conditions of the Leibniz test apply, and by that test, the series converges conditionally.

In Example 8.3.16, the Integral test is applied to the sequence $\left\{\ln \left(n\right)\&sol;n\right\}$, giving

${\int }_3^{\infty }\frac{\ln \left(x\right)}{x} \mathit{\&DifferentialD;}x$ = $\mathrm{\&infin;}$$$

Since this integral diverges, the given series does not converge absolutely.

The derivative is negative for $x\&gt;3$. (This is consistent with Figure 8.3.18(a).) Since the criteria of the test are satisfied, it follows that the series does not converge absolutely. The Leibniz test applies, and by it, this alternating series is seen to converge conditionally.