Q:

convergence of infinite series $\sum_{n=1}^\infty \frac{x^n}{(1+x)(1+x^2)(1+x^3)\cdot\cdot\cdot (1+ x^n)}$

I am reviewing for my final exam, and viewed this question:

Decide whether the following infinite sum is convergent for all $x >1$:
  $$\sum_{n=1}^\infty \frac{x^n}{(1+x)(1+x^2)(1+x^3)\cdot\cdot\cdot (1+
 x^n)}$$

I don't even know how to approach this question.  I have no idea what test should I try.  Any help would be highly appreciated!  Thanks

A:

Denote $u_n(x)$ the general term of the series then by the ratio test we have
$$\frac{u_{n+1}(x)}{u_n(x)}=\frac{x}{1+x^{n+1}}\sim_\infty\frac1{x^n}\xrightarrow{n\to\infty}0<1$$
so the given series is convergent for $x>1$.

