Lets say you want to know if $\sum_n^\infty sin(a_n)$ converges for some set of values $a_n$.
In many cases it might be really hard to prove (if not impossible) that $a_n\mod \pi$ eventually is not really close to integers. Even though a_n has no obvious relation to Pi. Then we might want to assume a_n mod Pi behaves somewhat randomly (but perfectly normal with regards to this property) and the divergence is trivial.

For instance $a_n=2^{1/3}*\zeta(2n+1)*e^{1/e}$  
Or $a_n=2^{1/2}*(the-n'th-prime-number)$