Product of random numbers. Find the constant $p$ such that the product of any (positive) number $N_0$ multiplied by successive random numbers between $0$ and $p$ will, on average, neither diverge to infinity nor converge to zero.
(This is not for homework or anything. This is a problem I "invented" and solved, and now want to see how real mathematicians would think about this. Take your time.)
(Thanks md2perpe for fixing up my tag.)
(Thanks Forester for making the variables render nicely.)

Thanks everyone. It turns out $p=e$ is exactly right, which all hinges on using the logarithm, and then transforming an infinite sum into an integral. My steps were pretty much the same:
Let $\tilde{N}$ denote the product after $j$ iterations. Then, we have $$\frac{\tilde{N}}{N_0} = \left(p \cdot r_1\right) \cdot \left(p \cdot r_2\right) \cdot \left(p \cdot r_3\right) \cdot \cdots \:,$$ where $p$ is constant, and each $r_j$ is a random number between $0$ and $1$. Next, take the natural log of both sides, turning the product on the right into sum:$$\ln\left(\frac{\tilde{N}}{N_0}\right) = \ln\left(p \cdot r_1\right) + \ln\left(p \cdot r_2\right) + \ln\left(p \cdot r_3\right) + \cdots$$
To satisfy the question as stated, we have the left side (on average) resolving to zero, with the number of terms on the right approaching infinity. This effectively makes $r$ a continuous variable: after many many iterations, we have sampled all of the real numbers between zero and one. Point is, we may replace the sum by an integral: $$0 \approx \ln\left(\frac{\tilde{N}}{N_0}\right) = \int_0^1 \left(\ln p + \ln r\right) dr \:,$$ and the answer falls right out. Only $p=e$ satisfies the above.
