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