Is $(|x_n|^{n+1})$ convergent or bounded if $|x_n|<1$? Let $(x_n)$ be a sequence such that $|x_n|<1$. Is $(|x_n|^{n+1})$ convergent or bounded? 

It is clearly bounded:
$$ |x_n|^{n+1} < 1^{n+1} = 1 \ . $$
It is not necessarily convergent. For example, take:
$$ x_n = \begin{cases} \left( \frac{1}{2} \right)^{\frac{1}{n+1}} & \mbox{ if  } n=2k \ , \\ \left( \frac{1}{3} \right)^{\frac{1}{n+1}} & \mbox{ if  } n=2k+1 \ . \end{cases} $$
