The 1/n variance rate requires uncorrelated samples

Let $X_1,\dots,X_n$ have common variance $\sigma^2$ and equicorrelation $\Cov(X_i,X_j) = \rho\sigma^2$ for $i\ne j$, with $\rho \in [0,1]$.

1. Compute $\Var(\bar X_n)$ exactly.
2. Find the limit as $n\to\infty$ and interpret it.
3. Define the effective sample size $n_{\text{eff}} = \sigma^2/\Var(\bar X_n)$ and compute it for $\rho = 0.1$ with $n = 10^6$.