=+3. In Problem 2 prove that the total variation inequality can be improved to S ZTV

=+3. In Problem 2 prove that the total variation inequality can be improved to

πS − πZTV ≤

2n+1 + e−1

(n + 1)! .

This obviously represents much faster convergence. (Hints: Use the exact probability of k matches p[k] from Example 4.3.1 and the second definition of the total variation norm in (7.6). Combine these with the binomial expansion of (1 + 1)n+1 and the bound

∞

j=k 1

j! ≤

2 k!

for k ≥ 1.)