I am trying to compute the inverval of convergence and the explicit value of the infinite series $$\prod_{n=1}^{\infty}\left(\dfrac{(n+1)x}{1+nx}\right).$$ I believe the interval of convergence is $(-1,1)$ and exact value is $\dfrac{x}{1-x},$ but I might be wrong. Any help would be greatly appreciated.
2026-04-02 19:20:37.1775157637
Values of the infinite product $\prod_n\frac{(n+1)x}{1+nx}$
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For numbers of the form $-\frac{1}{n}$ the product is not defined, for the other negatives values the product diverges to $\infty$ or $-\infty$. For $x \in [0,1)$ the product diverges to $0$. For $x>1$ the product diverges to $\infty$. So the product only converges for $x=1$.