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# Prove by induction a formula for the sum from 1 to ∞ of nkxn

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 