Four of the previous exercise (here, here, here, and here, Section 10.9, Exercises #11 – #14) suggest a formula
In the formula indicates a polynomial of degree , with the term of lowest degree being and the highest degree term being (we call a polynomial of degree with highest term a monic polynomial of degree ). Use mathematical induction to prove this formula, without justifying the manipulations with the series.
Proof. For the case , we have
from the first exercise linked above (Section 10.7, Exercise #11). Letting , a degree polynomial, with term of lowest degree being and term of highest degree being . Therefore, the statement holds for the case . Assume then that the statement is true for some , so
Differentiating both sides (assuming without justification that we can do this) we obtain
where is a polynomial of degree with the term of lowest degree being and that of the highest degree being . Hence, if the statement is true for then it is true for ; thus, the statement is true for all positive integers