Consider the sequence $(v_n)_{n \geq 0}$ defined in terms of the coefficients $u_n$ by the formula
$$v_n = n! \sum_{i=0}^{n} \frac{u_i}{i!}$$
and the power series
$$v(x) = \sum_{n=0}^{\infty} v_n x^n \in \mathbf{Q}\llbracket x \rrbracket.$$
Show the equality of power series
$$\sum_{n=0}^{\infty} (v_n - n v_{n-1}) x^n = \sum_{n=0}^{\infty} u_n x^n.$$