Answer
Conditionally convergent.
Work Step by Step
Alternating series Test states that if the following conditions are met, then the series is convergent.
1. $\lim\limits_{ n\to \infty} b_n=0$
2.$b_n$ is a decreasing sequence.
Consider $a_n=\Sigma_{n=1}^{\infty}\dfrac{(-1)^n}{ \ln(n+1)}$
From the given problem, we get $b_n=\dfrac{1}{\ln (n+1)}$
1. $\lim\limits_{ n\to \infty} b_n=\lim\limits_{ n\to \infty} \dfrac{1}{\ln (n+1)}=0$
2.$b_n=\dfrac{1}{\ln (n+1)}$ is a decreasing sequence.
Therefore, the given series is conditionally convergent.
