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title: Calculus B. Ⅱ Final Notes
tags: [Calculus]

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tags: Calculus
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# Calculus B. Ⅱ Final Notes

## Proposition 9.6.3

For a power series $\sum_{n=1}^\infty a_nx^n$, we have the following $3$ cases:

1. $\sum_{n=1}^\infty a_nx^n$ converges only at $0$. So the radius of convergence is $0$.
2. $\sum_{n=1}^\infty a_nx^n$ converges absolutely on $\mathbb{R}$. So the radius of convergence is $\infty$.
3. $\sum_{n=1}^\infty a_nx^n$ converges absolutely  $\forall x, |x|<r$ and diverges $\forall x, |x|>r$. So the radius of convergence is $r$.

### pf.

1. $\forall x\in\mathbb{R}\setminus\{0\}, \sum_{n=1}^\infty a_nx^n$ diverges. So the radius of is $0$.
2. $\forall x\in\mathbb{R}\setminus\{0\}, \sum_{n=1}^\infty a_nx^n$ converges. So by **Theorem 9.6.2**, $\sum_{n=1}^\infty a_nx^n$ converges absolutely on $\mathbb{R}$.
3. $\exists c\ne0\ni\sum_{n=1}^\infty a_nc^n$ converges and $\exists d\ne0\ni\sum_{n=1}^\infty a_nd^n$ diverges. By **Theorem 9.6.2**, $|c|\leq|d|$.
   First if $|c|=|d|=r>0$, by **Theorem 9.6.2**, $\sum_{n=1}^\infty a_nx^n$ converges absolutely $\forall x, |x|<|c|=r$ and diverges $\forall x, |x|>|d|=r$.
   So suppose $|c|<|d|$. Let $S=\{x|\sum_{n=1}^\infty a_nx^n\text{ converges absolutely}\}$.
   Then since $x\in S\forall x, |x|<|c|$, $S$ contains at least a positive number.
   Moreover, since $x\not\in S\forall x, |x|>|d|$, $S$ is bounded above.
   Thus by **Axiom 8.1.3**, $\exists\sup S=r>0$.
   By **Theorem 9.6.2** and **Theorem 8.1.4**, $|c|\leq r\leq|d|$.
   Finally by **Theorem 9.6.2**, $\sum_{n=1}^\infty a_nx^n$ converges absolutely $\forall x, |x|<|c|$ and diverges $\forall x, |x|>|d|$.
   Hence the radius is $r$.

## Theorem 11.4.2

If $\mathbf{x_0}$ is a local extreme point, then $\nabla f(\mathbf{x_0})=\mathbf{0}$ or $\nabla f(\mathbf{x_0})$ D.N.E.

### pf.

Let $f:D\mapsto\mathbb{R}, D\subseteq\mathbb{R}^2$ and $\mathbf{x_0}=(x_0,y_0)\in D$ is a local extreme point.

First if $\nabla f(\mathbf{x_0})$ D.N.E., then there's nothing further to prove.

So suppose $\nabla f(\mathbf{x_0})$ exists.

Let $g(x)=f(x,y_0)$. Then since $(x_0,y_0)$ is a local extreme point, $x_0$ is a local extreme point of $g$.

Then since $f$ is diff. at $(x_0,y_0)$, by **Theorem 11.2.3**, $f_x(x_0,y_0)$ exists. By **Theorem 3.3.2**, $g'(x_0)=f_x(x_0,y_0)=0$.

Similarly, $f_y(x_0,y_0)=0$.

So by **Theorem 11.2.3**, $\nabla f(x_0,y_0)=(f_x(x_0,y_0),f_y(x_0,y_0))=\mathbf{0}$

## 清華十大建設

1. 教育大樓
2. 美術大樓
3. 美齋
4. 大禮堂
5. 校門口
6. 文物館
7. 美術館
8. 文學館
9. 成功湖整治
10. 泰雅族館

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## Tylor Series

> To show a Tylor Series at $0$ converge to $f(x), \forall x\in\mathbb{R}$:

Since $f^{(n)}=$ ... _or_ $f^{(4m+1)}=$ ..., $f^{(4m+2)}=$ ..., $f^{(4m)}=$ ..., $f^{(4m+1)}=$ ...

Then by **Lagrange's Estimate**, $$0\leq|R_n(x)|\leq(\max_{t\in J}f^{(n+1)}(t)){|x|^{n+1}\over(n+1)!}\leq\dots$$

By **Pinching Theorem**, $\lim_{n\to\infty}R_n(x)=0$.

Hence $f(x)=\sum_{n=0}^\infty$...

## Power Series

> To find the interval of convergence $I$ of a power series $\sum_{n=1}^\infty a_nx^n$:

$$\lim_{n\to\infty}{a_{n+1}x^{n+1}\over a_nx^n}=L|x|$$

If $L=0$, $I=\mathbb{R}$. Else if $L$ doesn't exists, $I={0}$.

Otherwise the radius of convergence $r={1\over L}$.

If $x=r$, then $\sum_{n=1}^\infty a_n$ _converges or diverges?_

If $x=-r$, then $\sum_{n=1}^\infty(-1)^na_n$ _converges or diverges?_

Thus we have $I=$...

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## $\LaTeX$ & multiple-variable functions

- $f_x={\partial f\over\partial x}$
- $\nabla f(\mathbf{x})={\partial f\over\partial x}(\mathbf{x})\hat{\mathbf{i}}+\cdots$
  $\nabla f(\vec{x})={\partial f\over\partial x}(\vec{x})\hat{\imath}+\cdots$
- $f'_{\mathbf{u}}(\mathbf{x})=\nabla f(\mathbf{x})\cdot\mathbf{u}$
  $f'_{\vec{u}}(\vec{x})=\nabla f(\vec{x})\cdot\vec{u}$

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