I have a series
$$\sum_{r=1}^{\infty} \frac{e^{-rx}}{ln(r+1)}$$
Which converges for $x \geq 0$ and then I expand the exponential in the sum to get
$$\sum_{r=1}^{\infty} \frac{1}{ln(r+1)} \sum_{n=0}^{\infty} \frac{(-rx)^n}{n!}$$
And collect the r terms to get
$$\sum_{n=0}^{\infty} G_{n} \frac{x^{n}}{n!}$$ where $$G_{n}=\sum_{r=2}^{\infty} \frac{(-r)^n}{\ln {r}}$$
But $G_n$ is a divergent series right? Did I do it wrong or is it because of something else?