Convergence of CDF

Question

I have a cumulative probability distribution function F(x).

I need to prove that

$\sum_{n=2}^\infty n(F(x)^{n-1} - F(x)^n)$ does not converge for all $x \in \mathbb{R^+} - \{x:F(x)\ne 0\}$

Answer 1

That's wrong. For $F(x)<1$:$$\sum_{n=2}^\infty n(F(x)^{n-1} - F(x)^n)=(1-F(x))\sum_{n=2}^\infty nF(x)^{n-1}\\=(1-F(x))\dfrac{2F(x)-F^2(x)}{(1-F(x))^2}\\=F(x)\dfrac{2-F(x)}{1-F(x)}$$and for $F(x)=1$:$$\sum_{n=2}^\infty n(F(x)^{n-1} - F(x)^n)=0$$which is convergent at both cases.