I understand (a) and (c). I am more confused on how to prove it is unbounded. Also help on (d) and (e) would be great!
2026-04-01 12:13:46.1775045626
How to show that $P_n(x)$ is unbounded?
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By trivial induction $f^{(n)}(x) = (-1)^k \frac{k!}{x^{k+1}}$. Thus $P_n(x) = \sum_{k=0}^n (x-1)^k(-1)^k=\sum_{k=0}^n (1-x)^k$. For $x\neq 1$ this simplifies as $\frac{1-(1-x)^{n+1}}{x}$.