# Monte Carlo estimations of e

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.

• You would normally use MC to integrate something yielding $e$ in the end, no? – gt6989b Jan 28 '14 at 18:46
• You may want to check wiki.stat.ucla.edu/socr/index.php/… – Macavity Jan 28 '14 at 19:06
• A totally different idea: The probability that a random permutation of $n\gg 0$ objects is fixed-point free, is $\approx \frac1e$ – Hagen von Eitzen Jan 28 '14 at 19:54

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 $3^{1/r}$.