Difficult# Challenging Convergence

APCALC-UVKWG4

Which of the following series converge?

I.$\sum_{n=0}^{\infty }\cfrac{1\cdot 3\cdot 5\cdots ( 2n-1 )}{2\cdot 5\cdot 8\cdots ( 3n-1 )} $

II.$\sum_{n=0}^{\infty }\left ( \cfrac{4n}{8n-1} \right )^{n} $

III.$\sum_{n=0}^{\infty }\cfrac{2^{n}}{3^{n}} $