Monte Carlo estimations of e

Question

I need to estimate $e$ with a monte carlo method. We only learned the crude monte carlo integration, so I can't use any robust monte carlo simulations.

I know that $\displaystyle \int\limits_1^x \frac{1}{s}ds=\ln(x),$ so I just need to play around with the limit using randomly generated numbers.

Answer 1

The solution below assumes that we know how to take $r$-th roots, where $r$ is rational. I do not consider it a good solution.

Use the obvious Monte Carlo estimation of $\int_1^3 \frac{dx}{x}$ to estimate $\ln 3$. Call this estimate $r$.

Then $\ln 3\approx r$, and therefore $3\approx e^r$. Now calculate $3^{1/r}$.