Problem: Study the absolute and conditional convergence domain of the series: $$\sum_\limits{n=1}^{\infty}\frac{nx}{(1+x)(1+2x)...(1+nx)}$$ 1) for $0\leqslant x\leqslant \epsilon$,$\epsilon>0$ and 2) $\epsilon\leqslant x<+\infty$.
I chose to apply the Alambert critera:$\lim_{n\to\infty}\frac{\frac{(n+1)x}{(1+x)(1+2x)...(1+(n+1)x)}}{\frac{nx}{(1+x)(1+2x)...(1+nx)}}=\lim{n\to\infty}\frac{nx+x}{nx+n^2x^2+nx^2}=\lim_{n\to\infty}=\frac{x+\frac{x}{n}}{x+nx^2+x^2}=0$
Question:
Is my attempt right? Because if it is. Why would the author ask the question for different convergence domain("1) for $0\leqslant x\leqslant \epsilon$,$\epsilon>0$ and 2) $\epsilon\leqslant x<+\infty$")?
Thanks in advance!