You are asked to investigate the absolute convergence of the series $$\sum_{n=2}^{\infty} (-1)^{n+1} \left( 3+\frac{1}{n} \right) \frac{1}{\ln(n)}$$

This is absolutely convergent if and only if $$\sum_{n=2}^{\infty} \frac{1}{\ln(n)}$$ is absolutely convergent (explain why).

This last series can be compared to the harmonic series and doing so will allow you to conclude that it does not converge.

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