Let $\displaystyle a~[0], ... , a~[n]$ be real numbers such that $\displaystyle a~[0]$ is nonzero, and let $\displaystyle L$ be the following linear operator: $\displaystyle a~[n]D^n + ... + a~[1]D + a~[0]I$.

Let $\displaystyle p(t)$ be a polynomial. Let us define the functions $\displaystyle y~[0] = p(t)$. and: $\displaystyle y~[k+1] = p(t) - (a~[n]D^n + ... + a~[1]D)[y~[k]]$.

Prove that the functions $\displaystyle y~[k](t)$ converge to a polynomial $\displaystyle y(t)$, and that $\displaystyle L[y] = p(t)$