May 7th, 2016, 02:55 PM  #1 
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics  Is my proof correct?
This is actually a part of a multiplechoice question. I think I get the intuition behind it, but I'd like to make sure my proof is right as well. Thanks! For an integer $\displaystyle n \geq 0$, let $\displaystyle P_n(x)$ be the order n Taylor polynomial of a differentiable function $\displaystyle f(x)$ on $\displaystyle \mathbb{R}$ at $\displaystyle a \in \mathbb{R}$. Which of the following statements is/are true for every differentiable function $\displaystyle f$? I. For any given $\displaystyle x \in \mathbb{R}$, $\displaystyle f(x) = P_n(x)$ gets smaller as $\displaystyle n$ gets bigger. II. $\displaystyle f(a) = P_0(a)$ and $\displaystyle f^{(k)}(a) = P_n^{(k)}(a)$ for every $\displaystyle 1 \leq k \leq n$ III. $\displaystyle f(a) = P_0(a)$ and $\displaystyle f^{(k)}(a) = P_n^{(k)}(a)$ for every $\displaystyle k \geq 1$ For II, here's my work: Define $\displaystyle P_n(x) = \sum_{i = 0}^n \frac{f^{(i)}(a)}{i!} (xa)^i$ Then $\displaystyle \begin{align*} P_n'(x) &= \sum_{i = 1}^n \frac{f^{(i)}(a)}{i!} i(xa)^{i1}(10) + \frac{d}{dx}\frac{f^{(i)}(a)}{i!} (10)\\ &= \sum_{i = 1}^n \frac{f^{(i)}(a)}{i!} i(xa)^{i1}\\ P_n''(x) &= \sum_{i = 2}^n \frac{f^{(i)}(a)}{i!} i(i1)(xa)^{i2} \\ P_n^{(k)}(x) &= \sum_{i = k}^n \frac{f^{(i)}(a)}{i!} (\Pi_{j = i  k + 1}^i j) (xa)^{ik} \\ P_n^{(k)}(a) &= \sum_{i = k+1}^n \frac{f^{(i)}(a)}{i!} (\Pi_{j = i  k + 1}^i j) (aa)^{ik} + \frac{f^{(k)}(a)}{k!} (\Pi_{j = k  k + 1}^i j) (xa)^{kk} \\ &= \frac{f^{(k)}(a)}{k!} (k!)\\ &= f^{(k)}(a) \end{align*}$ For III, I only need to show it's false for k > n. Since $\displaystyle P_n(a)$ is a polynomial, $\displaystyle P_n^{(k)}(a) = 0$. But $\displaystyle f(x)$ may not be a polynomial, which implies $\displaystyle f^{(k)}(a) = 0$ isn't always true. Thus III is false. 
May 8th, 2016, 10:49 AM  #2 
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics 
Bump Nobody likes me 
May 8th, 2016, 02:58 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,854 Thanks: 744 
Looks good (I didn't check the details, but you have the right idea). There is a typo in I: should have a  sign, not an = sign. 
May 10th, 2016, 07:38 AM  #4  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902  Quote:
Quote:
 
May 10th, 2016, 09:00 AM  #5 
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics  

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correct, proof, taylor 
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