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Math Help - Taylor Polynomial Question

  1. #1
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    Taylor Polynomial Question

    Hi everyone,

    I was wondering if someone could give me a hand with the following problem.

    Suppose that p is a polynomial of degree d. Given any x0 inR, show that the (d+1)th Taylor polynomial for p at x0 is equal to p.

    What I've tried so far is basically the brute method. I tried setting up a general polynomial p of degree d and expanding and seeing what I came up with. As you can imagine, this hasn't been going so well . Any suggestions?

    Thanks in advance.


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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by seven.j View Post
    Hi everyone,

    I was wondering if someone could give me a hand with the following problem.

    Suppose that p is a polynomial of degree d. Given any x0 inR, show that the (d+1)th Taylor polynomial for p at x0 is equal to p.

    What I've tried so far is basically the brute method. I tried setting up a general polynomial p of degree d and expanding and seeing what I came up with. As you can imagine, this hasn't been going so well . Any suggestions?


    Thanks in advance.


    Tell me if this is too heavy duty:

    Spoiler:

    Let \text{deg}(p)=n fix x_0\in\mathbb{R} (where \mathbb{R}_n[x] is just the set of all n^{\text{th}}-degree polynomials of degree n) and let D_{x_0}^j:\mathbb{R}_n[x]\to\mathbb{R}_n[x],\quad j\in\{0,\cdots,n\} be the mapping \left(D_{x_0}^j(p)\right)(x)=(x-x_0)^j p^{(j)}(x_0). This mapping is linear since

    \begin{aligned}\left(D_{x_0}^j(\alpha p+\beta q)\right)(x) &= (x-x_0)^j\left(\alpha p+\beta q)^{(j)}(x)\\ &= (x-x_0)^j (\alpha p^{(j)}+\beta q^{(j)})(x_0)\\ &= (x-x_0)^j\left(\alpha p^{(j)}(x_0)+\beta q^{(j)}(x_0)\right)\\ &= \alpha (x-x_0)^j p^{(j)}(x_0)+\beta (x-x_0)^j q^{(j)}(x_0)\\ &= \alpha\left(D_{x_0}^j(p)\right)(x)+\beta\left(D_{x  _0}^j(q)\right)(x)\end{aligned}

    and thus the function


    T_n(p):\mathbb{R}_n[x]\to\mathbb{R}_n[x]


    where


    \displaystyle \left(T_n(p)\right)(x)=\sum_{j=0}^{n}\frac{1}{j!}D  _{x_0}^j(p)

    is linear. But, since \{1,\cdots,x^n\} is a basis for \mathbb{R}_n[x]; to prove that T_n(p)=p for all p\in\mathbb{R}_n[x] it suffices to check it for p(x)=x^j,\text{ }j=0,\cdots,n. To do this we merely note that


    \displaystyle \begin{aligned}T_n(x^m) &= \sum_{j=0}^{n}\frac{1}{j!}D^j_{x_0}(x^m)\\ &= \sum_{j=0}^{m}\frac{m(m-1)\cdots(m-j+1)}{j!}(x-x_0)^j x_0^{m-j}\\ &= \sum_{j=0}^{m}{m\choose j}(x-x_0)^j x_0^{m-j}\\ &= (x-x_0+x_0)^m\\ &=x^m\end{aligned}

    where we've used the fact that after the m^{\text{th}} term the terms vanish (that's how we can account for the change of the upper index) and at the last step we employed the Binomial theorem. The conclusion follows by noticing that T_n(p) is the n^{\text{th}} degree Taylor polynomial at x_0, where of course we realize that the remainder term is zero since the n+1^{\text{th}} derivative of p is identically zero.

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