If $\vert x \vert < 1$ then I want to show $\lim_{n\rightarrow \infty} (n+1)\cdot x^{n\cdot n!} < 1$
It makes perfectly sense in my world, because the factor $x^{n\cdot n!}$ is smaller than the factor $(n+1)$ when n goes to infinity. I have tried to use L'Hoptial but it doesn't work. Then I tried to find an example of an expression which is greater than $(n+1)\cdot x^{n\cdot n!}$ but still smaller than 1, when n goes to infinity. But all the examples I have found diverges. Now I'm stuck - any ideas?