User:Egm6341.s10.team3.ynahn81/HW1-2

= (2) Taylor Series Expansion of $$ f(x)=e^{x} $$= Ref: Lecture notes [[media:Egm6341.s10.mtg2.pdf|p.2-3]]

Problem Statement
$$\displaystyle f(x)=e^{x} $$

Find (i)$$\displaystyle P_{n}(x) $$ and

(ii)$$\displaystyle R_{n+1}(x) $$

Solution
From [[media:Egm6341.s10.mtg2.pdf|p.2-2]] and [[media:Egm6341.s10.mtg2.pdf|p.2-3]] of the class note, $$\displaystyle P_{n}(x) $$ and $$\displaystyle R_{n+1}(x) $$ are defined by
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P_{n}(x)=f(x_{0})+\frac{(x-x_{0})}{1!}f^{(1)}(x_{0})+\cdots+\frac{(x-x_{0})^{n}}{n!}f^{(n)}(x_{0}) $$ $$ and
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
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R_{n+1}(x)=\frac{1}{n!}\int_{x_{0}}^{x}(x-t)^{n}f^{(n+1)}(t)dt=\frac{(x-x_{0})^{n+1}}{(n+1)!}f^{(n+1)}(\xi), \quad \xi\in[x_{0},x] $$ $$ respectively.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
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Also, note that the following property of $$\displaystyle e^{x} $$:
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\frac{d^{n}}{dx^{n}}\left(e^{x}\right)=e^{x}, \quad n=1, 2, \cdots. $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 3)
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 * }

(i) Let's plug $$\displaystyle f(x)=e^{x} $$ into Eq. 1.
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P_{n}(x)=e^{x_{0}}+\frac{(x-x_{0})}{1!}\frac{d}{dx}\left(e^{x_{0}}\right)+\cdots+\frac{(x-x_{0})^{n}}{n!}\frac{d^{n}}{dx^{n}}\left(e^{x_{0}}\right) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 4)
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 * }

Now, use Eq. 3 to simplify Eq. 4. Then,
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$$\displaystyle \Rightarrow P_{n}(x)=e^{x_{0}}+\frac{(x-x_{0})}{1!}e^{x_{0}}+\cdots+\frac{(x-x_{0})^{n}}{n!}e^{x_{0}} $$ $$
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 * $$\displaystyle (Eq. 5)
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(ii) Let's plug $$\displaystyle f(x)=e^{x} $$ into Eq. 2.
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R_{n+1}(x)=\frac{1}{n!}\int_{x_{0}}^{x}(x-t)^{n}\frac{d^{n+1}}{dt^{n+1}}\left(e^{t}\right)dt=\frac{(x-x_{0})^{n+1}}{(n+1)!}\frac{d^{n+1}}{dx^{n+1}}\left(e^{x}\right), \quad x= \xi, \quad \xi\in[x_{0},x] $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 6)
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Now, use Eq. 3 to simplify Eq. 6. Then,
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$$\displaystyle \Rightarrow R_{n+1}(x)=\frac{1}{n!}\int_{x_{0}}^{x}(x-t)^{n}e^{t}dt=\frac{(x-x_{0})^{n+1}}{(n+1)!}e^{\xi}, \quad \xi\in[x_{0},x] $$ $$
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 * $$\displaystyle (Eq. 7)
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Hence $$ \displaystyle f(x)$$ can be expressed as follows: $$\displaystyle f(x)=e^{x} $$ can be expressed in Taylor series expansion by
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$$\displaystyle f(x)=e^{x}=\underbrace{e^{x_{0}}\sum_{j=0}^{n}\frac{(x-x_{0})^{j}}{j!}}_{P_{n}(x)}+\underbrace{\frac{(x-x_{0})^{n+1}}{(n+1)!}e^{\xi}}_{R_{n+1}(x)}, \quad \xi\in[x_{0},x] $$ $$
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 * $$\displaystyle (Eq. 8)
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