Given $S=\sum_{i=0}^{i=n}{(-1)^i\binom{n}{i}}P_k(i)$ where $P_k(i)$ is a polynomial of degree $k$ in $i$. I have to prove that $S=0$ for $n>k$. How can I do that?
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First show that $\sum_{i=0}^n (-1)^i \binom{n}{i} i^k=0$: https://math.stackexchange.com/questions/3833276/if-px-is-any-polynomial-of-degree-less-than-n-show-that-sum-j-0n-1/3833675#3833675 – RobPratt Oct 12 '20 at 21:00
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3Does this answer your question? Summation of series involving binomial coefficients and polynomial of degree at most n-1 – RobPratt Oct 12 '20 at 21:02
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yes now i see that my question is duplicated! Many thanks! – maximiliano1 Oct 15 '20 at 14:04