Interpolation, proving P[x_0,x_1, ,x_n] = 0

[peripatein]

Hi,

Homework Statement

I am asked to show that P[x₀,x₁,...,x_n] = 0 for P(x) [itex]\in[/itex]∏_n-1 without directly using the property that the divided difference
over n+1 nodes is the nth derivative. The hint says to use the formula for the error.

Homework Equations

The Attempt at a Solution

I am not quite sure how to approach this. I am familiar with the expression for the error. I am also familiar with Rolle's Theorem, or Mean Value Theorem, but am I allowed/supposed to use that based on the indications in the question?
I'd appreciate some guidance. Thanks!

 

[KataKoniK]

Hello,

Thank you for your question. It seems like you are trying to prove that the divided difference over n nodes is equal to the nth derivative without directly using that property. One approach you could take is to use Taylor's theorem to show that the error term in the divided difference formula is equal to 0. This would then imply that the divided difference itself is equal to 0.

Another approach could be to use mathematical induction to show that the divided difference over n nodes is equal to the nth derivative, starting with n=1 and then showing that if it holds for n, it also holds for n+1. This would indirectly prove the desired property without explicitly using it.

I hope this helps. Good luck with your proof!