Infinite series : $\sum_{n=1}^{\infty}\prod_{i=1}^n\frac{(3i-1)}{(4i-3)} $ How to evaluate this?
$$\sum_{n=1}^{\infty}\prod_{i=1}^n\frac{(3i-1)}{(4i-3)} $$

Nope it doesn't diverge the limit is $2 \cdot \text{Hypergeometric2F1}[1,\frac{5}{3},\frac{5}{4},\frac{3}{4}]$  (according to Mathematica)
I Interpret it as 
$ \sum_{n=1}^\infty \prod_{i=1}^n \frac{3i-1}{4i-3}$
