Question Number 106644 by ZiYangLee last updated on 06/Aug/20
$$\mathrm{Let}\:\mathrm{P}\left(\mathrm{x}\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{of}\:\mathrm{degree}\:\mathrm{n} \\ $$$$\mathrm{with}\:\mathrm{real}\:\mathrm{coefficients}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{P}^{\left(\mathrm{k}\right)} \left(\mathrm{0}\right)}{\left(\mathrm{k}+\mathrm{1}\right)!}=\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} \mathrm{P}^{\left(\mathrm{k}\right)} \left(\mathrm{1}\right)}{\left(\mathrm{k}+\mathrm{1}\right)!} \\ $$$$ \\ $$