sequences and series - Is $sum_{n=1}^{+infty}frac{(-1)^n log n}{n!}$ a positive sum?

The below series is convergent series by the ratio test but i'm no able to know if this series have a positive sum , and i don't succeed to check if it has a closed form ,Then my question here is :

Question:
Is this : $\displaystyle\sum_{n=1}^{+\infty}\frac{(-1)^n \log n}{n!}$ a positive sum ?

Answer

For $n \geq 2$, the expression $\frac{\log n}{n!}$ is strictly decreasing, as $\frac{\log (n+1)}{(n+1)!} < \frac{\log n}{n!} \iff \log (n+1) < (n+1) \log n \iff \log_n (n+1) < n+1$, which is true, because $\log_n (n+1) < \log_n (n^2) = 2$. Therefore, $$\sum_{n=1}^\infty \frac{(-1)^n \log n}{n!}$$ is an alternating series with terms that strictly decrease in magnitude after the first nonzero term, so the series has the same sign as the first nonzero term, which is positive.