Numerical Methods: Euler to Runge–Kutta · Superchallenge problem

The leapfrog method $y_{n+1} = y_{n-1} + 2h\,f(t_n, y_n)$ is applied to $y' = -y$ with $h = \dfrac{1}{n}$. Its general solution is $y_k = A\xi_1^{\,k} + B\xi_2^{\,k}$, where $\xi_1 > 0$ is the root of the characteristic equation that approximates $e^{-h}$ and $\xi_2$ is the other (parasitic) root. Find $\displaystyle\lim_{n \to \infty} \dfrac{|\xi_2|^n}{\xi_1^{\,n}}$, the factor by which the parasitic mode outgrows the true one over the time interval $[0, 1]$.