Prove $\sum_{i=0}^{n}\left(x_{i}^{n}\prod_{0\leq k\leq n}^{k\neq i}\frac{x-x_k}{x_i-x_k}\right)=x^n$

Question

Suppose $x_0$ , $x_1$ , $x_2$ , ... , $x_n$ are distinct real numbers , prove that :

$$ \large{\displaystyle{\sum_{i=0}^{n} \left( x_{i}^{n}\prod_{\substack{0\leq k\leq n \\ k\neq i }}\frac{x-x_k}{x_i-x_k} \right)=x^n}} $$

I have no ideas to do this question

Both sides are polynomials of degree $\le n$ in $x$. Hence they are equal, if they are equal at $n+1$ disticnt points. Check equality at $x = x_j$, $0 \le j \le n$. — martini, Oct 24 '12 at 11:58

score 8 · Accepted Answer · answered Oct 24 '12 at 12:30

8

That's simply Lagrange's polynomial interpolation formula for the values of the polynomial $x^n$. Since there are $n+1$ data points, the two polynomials coincide.

answered Oct 24 '12 at 12:30

lhf

221,500

also re. to answer to http://math.stackexchange.com/questions/1836535/prove-lagrange-form-of-interpolation-for-xj – G Cab Jul 08 '16 at 15:41

Prove $\sum_{i=0}^{n}\left(x_{i}^{n}\prod_{0\leq k\leq n}^{k\neq i}\frac{x-x_k}{x_i-x_k}\right)=x^n$

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