User:Egm6341.s10.team3.Min/HW1-5

= Problem 5: Show Taylor series by integration =

Given

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$$ $$
 * $$\displaystyle f(x) = f(x_0) + \frac{(x-x_0)}{1!}\ f^{(1)}(x_0) + \int_{x_0}^{x} (x-t) f^{(2)}(t)\, dt
 * $$\displaystyle f(x) = f(x_0) + \frac{(x-x_0)}{1!}\ f^{(1)}(x_0) + \int_{x_0}^{x} (x-t) f^{(2)}(t)\, dt
 * $$\displaystyle (Eq. 1)
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Part 1
Repeat int. by parts to reveal the following items with remainder


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 * $$\displaystyle \frac{(x-x_0)^2}{2!}\ f^{(2)}(x_0) + \frac{(x-x_0)^3}{3!}\ f^{(3)}(x_0) $$


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Part 2
Assume (4)and (5) in [[media:egm6341.s10.p2-2.png|Lecture p.2-2]] are true, do int. by parts one more time

S 1
Int.$$\int_{x_0}^{x} (x-t){f^{(2)}(t)} dt $$ by parts in Eq 1:


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\int_{x_0}^{x} \underbrace{(x-t)}_{U^\prime} \underbrace{f^{(2)}(t)}_{V}\, dt & = \left [ UV \right ]_{x_0}^x -  \int_{t=x} UV^\prime\, dt \\ & = \left [ -\frac{(x-t)^2}{2}\ f^{(2)}(t) \right ]_{t=x_0}^{t=x} + \underbrace{\int_{x_0}^{x} \frac{(x-t)^2}{2}f^{(3)}(t)\, dt}_{W1} \\ & = \cancelto{o}{-\frac{(x-x)^2}{2}\ f^{(2)}(x)} + \frac{(x-x_0)^2}{2}\ f^{(2)}(x_0) + W1\\ & = \frac{(x-x_0)^2}{2!}\ f^{(2)}(x_0) + W1\\ \end{align} $$ $$
 * $$ \displaystyle \begin{align}
 * $$ \displaystyle \begin{align}
 * $$\displaystyle (Eq. 2)
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Use Eq 1, Eq 2, we have:
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$$ $$
 * $$\Rightarrow \ f(x) = f(x_0) + \frac{(x-x_0)}{1!}\ f^{(1)}(x_0) + \frac{(x-x_0)^2}{2!}\ f^{(2)}(x_0) + \frac{1}{2!}\int_{x_0}^{x} {(x-t)^2}f^{(3)}(t)\, dt
 * $$\Rightarrow \ f(x) = f(x_0) + \frac{(x-x_0)}{1!}\ f^{(1)}(x_0) + \frac{(x-x_0)^2}{2!}\ f^{(2)}(x_0) + \frac{1}{2!}\int_{x_0}^{x} {(x-t)^2}f^{(3)}(t)\, dt
 * $$\displaystyle (Eq. 3)
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Int. $$\int_{x_0}^{x} {(x-t)^2}f^{(3)}(t)\,dt $$ by part one more time in Eq 3:
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\int_{x_0}^{x} \underbrace{(x-t)^2}_{U^\prime} \underbrace{f^{(3)}(t)}_{V}\, dt & = \left [ UV \right ]_{x_0}^x -  \int_{t=x} UV^\prime\, dt \\ & = \left [ -\frac{(x-t)^3}{3}\ f^{(3)}(t) \right ]_{t=x_0}^{t=x} + \underbrace{\int_{x_0}^{x} \frac{(x-t)^3}{3}f^{(4)}(t)\, dt}_{W2} \\ & = \cancelto{o}{-\frac{(x-x)^3}{3}\ f^{(3)}(x)} + \frac{(x-x_0)^3}{3}\ f^{(3)}(x_0) + W2\\ & = \frac{(x-x_0)^3}{3}\ f^{(3)}(x_0) + W2\\ \end{align} $$ $$ Use Eq 3, Eq 4, we have:
 * $$ \displaystyle \begin{align}
 * $$ \displaystyle \begin{align}
 * $$\displaystyle (Eq. 4)
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$$\Rightarrow \ f(x) = f(x_0) + \frac{(x-x_0)}{1!}\ f^{(1)}(x_0) + \frac{(x-x_0)^2}{2!}\ f^{(2)}(x_0) + \frac{(x-x_0)^3}{3!}\ f^{(3)}(x_0) + \underbrace{\frac{1}{3!}\int_{x_0}^{x} (x-t)^3f^{(4)}(t)\, dt}_{Remainder} $$
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S 2
Int. $$\int_{x_0}^{x} {(x-t)^n}f^{(n+1)}(t)\,dt $$ by part one more time:


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\int_{x_0}^{x} \underbrace{(x-t)^n}_{U^\prime} \underbrace{f^{(n+1)}(t)}_{V}\, dt & = \left [ UV \right ]_{x_0}^x -  \int_{t=x} UV^\prime\, dt \\ & = \left [ -\frac{(x-t)^{n+1}}{n+1}\ f^{(n+1)}(t) \right ]_{t=x_0}^{t=x} + \underbrace{\int_{x_0}^{x} \frac{(x-t)^{n+1}}{n+1}f^{(n+2)}(t)\, dt}_{W} \\ & = \cancelto{o}{-\frac{(x-x)^{n+1}}{n+1}\ f^{(n+1)}(x)} + \frac{(x-x_0)^{n+1}}{n+1}\ f^{(n+1)}(x_0) + W\\ & = \frac{(x-x_0)^{n+1}}{n+1}\ f^{(n+1)}(x_0) + W\\ \end{align} $$ $$
 * $$ \displaystyle \begin{align}
 * $$ \displaystyle \begin{align}
 * $$\displaystyle (Eq. 5)
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Use Eq 5, (4)& (5) in [[media:egm6341.s10.p2-2.png|Lecture p.2-2]], we have:
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$$\displaystyle \Rightarrow \ f(x) = \underbrace{f(x_0) + \frac{(x-x_0)}{1!}\ f^{(1)}(x_0) + \cdots + \frac{(x-x_0)^{n+1}}{n+1}\ f^{(n+1)}(x_0)}_{P_{n+1}(x)} + \underbrace{\frac{1}{(n+1)!}\int_{x_0}^{x} (x-t)^{n+1}f^{(n+2)}(t)\, dt}_{R_{n+2}(x)} $$
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