Let-P-x-be-a-polynomial-of-degree-n-with-real-coefficients-Prove-that-k-0-n-P-k-0-k-1-k-0-n-1-k-P-k-1-k-1-

Question Number 106644 by ZiYangLee last updated on 06/Aug/20

Let P(x) be a polynomial of degree n with real coefficients. Prove that Σ_(k=0) ^n ((P^((k)) (0))/((k+1)!))=Σ_(k=0) ^n (((−1)^k P^((k)) (1))/((k+1)!))

$$\mathrm{Let}\:\mathrm{P}\left(\mathrm{x}\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{of}\:\mathrm{degree}\:\mathrm{n} \\ $$$$\mathrm{with}\:\mathrm{real}\:\mathrm{coefficients}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{P}^{\left(\mathrm{k}\right)} \left(\mathrm{0}\right)}{\left(\mathrm{k}+\mathrm{1}\right)!}=\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} \mathrm{P}^{\left(\mathrm{k}\right)} \left(\mathrm{1}\right)}{\left(\mathrm{k}+\mathrm{1}\right)!} \\ $$$$ \\ $$

Facebook Tweet Pin

Let-P-x-be-a-polynomial-of-degree-n-with-real-coefficients-Prove-that-k-0-n-P-k-0-k-1-k-0-n-1-k-P-k-1-k-1-

Leave a Reply Cancel reply