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

You can sum this in terms of the hypergeometric function. Here is a result by maple
$$ 2\,{ _2F_1\left( 1, \frac{5}{3}; \,\frac{5}{4};\,\frac{3}{4}\right)}. $$
