I am trying to show that if $u_{0}=u_{1}=u_{2}=0$, and if, when $n>1$, $u_{2n-1}=\dfrac {-1} {\sqrt {n}}, u_{2n}=\dfrac {1} {\sqrt {n}}+\dfrac {1} {n}+\dfrac {1} {n\sqrt {n}}$ then $\prod \limits_{n=0}^{\infty }\left( 1+u_{n}\right) $ converges.
We observe that $\sum \limits_{n=0}^{\infty }u_{n}$ and $\sum \limits_{n=0}^{\infty }u_{n}^{2}$ are divergent by ratio test. I am unsure how to proceed from here. Any help would be much appreciated. Could we argue possibly since $\lim _{n\rightarrow \infty }u_{n}=0$ that's why $\prod \limits_{n=0}^{\infty }\left( 1+u_{n}\right) $ converges ?