Unfortunately I'm not talking about Taylor's theorem, but Taylor's polynomial. The problem stated is : Let be a function that can be derived at least times in . Define with precision what is the Taylor's polynomial of order of in . Prove that it exists and that it is unique.

My attempt : I define it : The Taylor's polynomial of order of in is the only polynomial of degree lesser or equal to that satisfies , , with union . Now I have to prove that it exists and that it's unique. How can I prove that?

Looking to my notes, the proof is not done, but it says to write the polynomial this way : Let the polynomial be , with and then after a few algebra with the derivatives of the polynomial, it says . The funny part is that now it says, ) shows the existence while ) shows the uniqueness.

I'm very confused. If I understood well, I must show thatifthe polynomial can be written such as above,thenand thatif,thenthe polynomial can be written as above. But I'm not sure, so I'm asking you. And in case of an affirmation, would you help me? I thought about induction but realized it was not worth it.