Does this imaginary series really diverge?

Question

$$\sum_{n=2}^{\infty}{\frac{(-i)^{n}}{\ln n}}$$ In the answer, using comparison test $1/\ln n > 1/n,$ the series is divergent. But, in my opinion, the series can be divided like $$\sum_{n=1}^{\infty}{\frac{(-1)^{n}}{\ln (2n)}}+i\sum_{n=1}^{\infty}{\frac{(-1)^{n+1}}{\ln (2n+1)}}$$ Then, each series converges, so the series should converge. What is the correct answer?

Answer 1

$a_n=\frac 1{\ln n}$ is monotically tends to zero. For $b_n=(-i)^n$ we have $|\sum_{n=2}^N b_n|\leq\sqrt2$ for any $N\geq 2.$ So, according to Dirichlet Test, $$\sum_{n=2}^\infty a_nb_n=\sum_{n=2}^\infty\frac{(-i)^n}{\ln n}$$ converges.