Ok so are IID, Uniform on [0,c]. And
Well the expectation of the sum is the sum of the expectations. And the expectation of the product, if independent, is the product of the expectations. So
Which is a geometric series (minus the zeroth term). We know that converges for ...?
As for the second part, ask yourself this question. If the expectation is finite, could the probability of being infinite be non-zero? So doesn't the answer to part 1 answer part 2?