Verify the relation, for all non-zero natural number $n$, $$2 d_{VT}\left(p_{X_n}, \pi_1\right) = \sum_{k=0}^{n} \frac{1}{k!} \left|\sum_{i=n-k+1}^{+\infty} \frac{(-1)^i}{i!}\right| + e^{-1} \sum_{k=n+1}^{+\infty} \frac{1}{k!}.$$