Avalanche Limits

Relevant wiki: Riemann Sums

$\begin{aligned} L & = \lim_{n \to \infty} \prod_{k=1}^{2n} \ln \left(1+\frac kn\right)^\frac 1n \\ & = \lim_{n \to \infty} \frac 1n \sum_{k=1}^{2n} \ln \left(1+\frac kn\right) & \small \color{#3D99F6} \text{Using Riemann sums} \\ & = \int_0^2 \ln (1+x) \ dx & \small \color{#3D99F6} \lim_{n \to \infty} \frac 1n \sum_{k=a}^b f \left(\frac kn\right) = \lim_{n \to \infty} \int_\frac an^\frac bn f(x) \ dx \\ & = (1+x)\ln(1+x) - x \ \bigg|_0^2 \\ & = 3\ln 3 - 2 = \ln 27 - 2 \end{aligned}$

Therefore, $a+b = 27+2 = \boxed{29}$ .