Calculus B. Ⅱ Final Notes

--- tags: Calculus --- {%hackmd BkVfcTxlQ %} # 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 --- ## 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 $I=$... --- ## $\LaTeX$ & multiple-variable functions - $f_x={\partial f\over\partial x}$ - $\nabla f(\mathbf{x})={\partial f\over\partial x}(\mathbf{x})\hat{\mathbf{i}}+\cdots$ $\nabla f(\vec{x})={\partial f\over\partial x}(\vec{x})\hat{\imath}+\cdots$ - $f'_{\mathbf{u}}(\mathbf{x})=\nabla f(\mathbf{x})\cdot\mathbf{u}$ $f'_{\vec{u}}(\vec{x})=\nabla f(\vec{x})\cdot\vec{u}$ {%hackmd @nevikw39/signature %}