Problem Solve for Mathematical Analysis II final
The scope of the midterm exam is Inverse function theorem, and in this part, understanding the proof seems more important than the problem.
However, the scope of the final exam seems important to apply the theory to a variety of examples, and it applies to mathematical physics.
35. (Limit comparison test) Given two positive sequences $\{a_j\}^\infty_{j=1}$ and $\{b_j\}^\infty_{j=1}$, suppose that the limit $\lim_{j\rightarrow \infty} \frac{a_j}{b_j}=L$ exists, and that the constant $L$ is positive. By usingthe completeness of the space $(\mathbb{R},\left| \cdot\right|)$, show that the series $\sum a_j$ converges if and only is the series $\sum b_j$ converges.
Answer:
$\lim_{\j\rightarrow \infty} \frac{a_j}{b_j}=L $ exists means $\forall \epsilon > 0,\ \exists N \in \mathbb{N}$ s.t. $\left| \frac{a_j}{b_j} - L\right| <\epsilon$ for all $n>N$.
Then $-\epsilon -L < \frac{a_j}{b_j} < \epsilon + L$.
Let $A_n=\sum_{j=1}^n a_j$, $B_n=\sum_{j=1}^n b_j$
$\left| A_n-A_m\right| = \left| \sum_{j=m+1}^n a_j\right| = \sum_{j=m+1}^n \left| a_j\right| < \sum_{j=m+1}^n (\epsilon+L)\left| b_j\right|. Since $\sum b_j converge, $\sum a_j$ converge.
$\blacksquare$
36. (A variation of the root test) Given a positive sequence $\{a_j\}^\infty_{j=1}$, show that if $$\limsup_{j\rightarrow \infty} \frac{a_{j+1}}{a_j}<1$$ then the series $\sum a_j$ converges.
Answer:
$\forall \epsilon,\ \exists n_0$ s.t. $\forall j \ge n_0$ $\frac{a_{j+1}}{a_j} < L+\epsilon$, $a_{j+1}<(L+\epsilon)a_j$
Smaller than geometric series $\blacksquare$
37. (Another variation of the root test) GIven a positive sequence $\{a_j\}^\infty_{j=1}$, show that if $$\limsup_{j\rightarrow \infty} \sqrt[j]{a_j} > 1$$, then the series $\sum a_j$ diverges, that is , tha limit $\lim_{k\rightarrow \infty} \sum_{j=1}^k a_j$ Does not exists.
38. (Another application of the summation by parts) Given two real $\{a_j\}^\infty_{j=1}$ and $\{b_j\}^\infty_{j=1}$, consider a series of the form $$\sum_{j=1}^\infty a_j b_j$$
Set $S_k := \sum_{j=1}^k a_jb_j$ and $A_k:=\sum_{j=1}^k a_j$ with $A_0=0$. Then, $$S_k=A_kb_{k+1}-\sum_{j=1}^kA_j(b_{j+1}-b_j)$$
Suppose that the sequence $\{A_j\}^\infty_{j=1}$ is bounded, and that $\{b_j\}^\infty_{j=1}$ is nonnegative and monotonially converges to $0$. Show that the seires $\sum a_j b_j$ ($=\lim_{k\rightarrow \infty} S_k$) converges.
Answer:
$\left| s_m - s_n\right| = \left| A_mb_{m+1} - A_n b_{n+1} - \sum_{j=n+1}^m A_k(b_{j+1}-b_j)\right| \le Mb_{m+1}+Mb_{n+1}+\sum M(b_{j+1}-b_j)$ = 2Mb_{m+1}$
$\blacksquare$
39. Given two positive sequence $\{a_j\}$ and $\{b_j\}$, suppose that $$\frac{a_{j+1}}{a_j}\le \frac{b_{j+1}}{b_j} \mbox{ fot all } j\in \mathbb{N}$$
Show that if $\sum b_j$ converges, then $\sum a_j$ converges.
Answer:
Multiply $j=1$ to $n-1$, $\frac{a_n}{a_1}\le \frac{b_n}{b_1}$
$\left| A_m-A_n\right| = \left| \sum_{j=n+1}^m a_j\right| \le \left| \sum \frac{a_1}{b_1}b_j\right|$...
$\blacksquare$
40. Given a sequence $\{x_j\}$ in a Banach space $(\mathbf{V},||\cdot||)$, define $$T_k:=\sum_{j=1}^k ||x_j||\mbox{ for } k\in \mathbb{N}$$
Show that if the sequence $\{T_k\}$ is bounded, then the series $\sum x_j$ converges in $(\mathbf{V},||\cdot||)$.
Answer:
Let $S_n=\sum_{j=1}^n x_j$. Then $\left| S_n-S_m\right| = \left| \sum x_j\right| \le \sum ||x_j|| \le \left| T_n -T_m\right| <\epsilon$
$T_k$ is bounded, monotonically increaing sequence.
Then it converge.
$\blacksquare$
41. For any real number $x\in \mathbb{R}\backslash \{2m\pi : m\in \mathbb{Z}\}$, prove the formula
$$ \sum_{j=1}^k \cos(jx) = \frac{\cos [(k+1)x/2] \sin (kx/2)}{\sin (x/2)}$$
$$\sum_{j=1}^k \sin(jx) = \frac{\sin [(k+1)x/2]\sin(kx/2)}{\sin(x/2)}$$
Answer:
use $e^{ijx}$, then just gemetric sum
$\blacksquare$
42. Fix a real number $x\in \mathbb{R} \backslash \{2m\pi: m\in \mathbb{Z}\}$. Let $\{a_j \}$ be a positive sequence that converges monotonically to $0$. Show that the series $\sum a_j \sin(jx) $converges.
Answer:
Use above euqation, $\sum sin(jx)$ bounded. By 38, it converge.
$\blacksquare$
43. Given a strictly posirive sequence $\{a_j\}$, that is, $a_j>0$ for all $j\in \mathbb{N}$, suppose that $\sum a_j$ diverges. Show that $\sum \frac{a_j}{1+a_j}$ diverges.
Answer:
use $\frac{a_j}{1+a_j} > \frac{a_j}{2}$ and comaprison.
$\blacksquare$
44. Given a nonnegative sequence $\{a_j\}$, that is , $a_j \ge 0$ for all $j\in \mathbb{N}$, show that if $\sum a_j$ converges, then $\sum \frac{\sqrt{a_j}}{j}$ converges as well.
Answer:
use $(a_j + \frac{1}{j^2}) \ge 2\sqrt{a_j\frac{1}{j^2}}$.
45. Fix a constant $c>0$. And, fix another constant $d>0$ such that $$d>\max \{1,c\}$
Show that the series $$\sum_{j=1}^\infty \frac{1}{c^j-d^j}$$ converges.
Answer:
use $\frac{1}{c^j-d^j} < \frac{1}{d^j}$ $\blacksquare$
46. Prove the following statement:
Theorem. Let $\sum a_j$ be a series of real numbers which converges absolutely. For any rearrangement $\sum a_{\phi(j)}$, it holds that $$\sum a_j = \sum a_{\phi(j)}$$
Answer:
Select bigger series that include before series.
47. The series $$\sum_{j=1}^\infty \frac{(-1)^{j+1}}{\sqrt{j}}$$ converges conditionally. Show that the rearrangement $$1+\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{5}}+\frac{1}{\sqrt{7}}-\frac{1}{\sqrt{4}}+\frac{1}{\sqrt{9}}+\frac{1}{\sqrt{11}}-\frac{1}{\sqrt{6}}+\frac{1}{\sqrt{13}}+\frac{1}{\sqrt{15}}-\frac{1}{\sqrt{8}}\cdots$$
diverges.
48. Define $$l_2 := \{ \{x_j\}_{j=1}^\infty:x_j\in \mathbb{R}\ \forall j \in \mathbb{N},\ \sum_{j=1}^\infty x_j^2 < \infty\}$$
(a) Show that $l_2$ is a real vector space.
(b) For $x=\{x_j\}_{j=1}^\infty,\ y=\{y_j\}_{j=1}^\infty \in l_2$, define $$ \left\langle x,y\right\rangle := \sum_{j=1}^\infty x_jy_j$$
Show that $\left\langle \cdot,\cdot\right\rangle$ defines an inner product on $l_2$.
(c) Prove that Cauchy-Scwarz inequality in $l_2$: For $x=\{x_j\}_{j=1}^\infty,\ y=\{y_j\}_{j=1}^\infty \in l_2$, $$\left| \left\langle x,y\right\rangle \right| \le ||x||||y||$$
(d) Definee a metric $d$ on $l_2$ by $$ d(x,y):=||x-y||$$
Show that $(l_2,d)$ is a Banach space.
49. Show that the series $$\sum_{k=2}^\infty ((-1)^k\sum_{j=2^{k-1}}^{2^k-1} \frac{1}{j\ln j})$$ converges.
Answer:
Calculate $\int_{2^{k-1}}^{2^k} \frac{1}{x\ln x} dx$ and sqeeze. $\blacksquare$
50. Definition. Given two real sequences $\{a_j\}_{j=0}^\infty$ and $\{b_j\}_{j=0}^\infty$, define $$c_k:=\sum_{j=0}^k a_j b_{k-j}$$
(i) The sequence $\{F_n := \sum_{j=1}^n f_j\}$ is uniformly bounded, that is, there exists a finite constant such that $$\left| F_n(x)\right| \le M\quad \mbox{for all } x\in \Omega,\ n \in \mathbb{N}$$
(ii) The sequence $\{g_j\}$ is monotone (either monotonically increasing, and uniformly converges to $0$ on $\Omega$.
Show that the series $\sum_j f_j g_j$ converges uniformly on $\Omega$.
56. Show that the series $$\sum_{j=0}^\infty \frac{2x}{j^2-x^2}$$ converges uniformly on any interval $[a,b] that contains no interges.
57. Evaluate $$\int_0^{\frac{\pi}{2}} \sum_{j=1}^\infty \frac{\sin((2j+1)x)}{2j(2j-1)}dx$$
58. Suppose that $\sum_{j=0}^\infty a_jx^j$ converges to $f(x)$ on $(-R_1, R_1)$ and that $\sum_{j=0}^\infty b_jx^j$ converges to $g(x)$ on $(-R_2,R_2)$. Define $R$ by $$R:= \min \{R_1, R_2\}$$
Prove that the Cauchy product of $\sum_{j=0}^\infty a_jx^j$ and $\sum_{j=0}^\infty b_jx^j$ converges to $f(x)g(x)$ on $(-R,R)$.
59. Use the power series expansion to $\sin x$ about $x_0=0$ to evaluate the integral $$\int_0^x \frac{\sin t}{t} dt$$
60. (a) Find the power series represetation about $x_0=0$ of $\frac{1}{1+x^2}$, and find the radius of convergence.
(b) By using the result obtained from (a), find the power series representation about $x_0=0$ of $\tan^{-1} x$. Show that this power series representation is valid for $-1<x\le 1$.
(c) By susing the result obtained from (b), evaluate the integral $$\int_0^x \arctan t dt$$ as a power series about $x_0$ for $-1<x\le 1$.
(d) Prove that $$\frac{\pi}{4} -\ln \sqrt{2}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}\cdots$$
61. Find the radius of convergence of the series $$\sum_{j=0}^\infty \frac{(2j)!}{(j!)^2}x^j$$
62. For a function $f(x)$ defined by $$f(x)=\sum_{j=1}^\infty \frac{(-1)^j}{2j} x^{2j+1}\quad \mbox{for } -1\le x \le 1$$ show that $f$ is differentiable for all $x\in (-1,1)$.
63. Prove the following statement.
Theorem. If the radius of convergence of a power series $P(x)=\sum_{j=0}^\infty c_j(x-x_0)^j$ is $R$, and if $P(x)$ converges at $x=x_0-R$, then, for any $\epsilon \in (0,R)$, the power series $P(x)$ converges uniformly on $[x_0-R,x_0+R-\epsilon]$.
64. Show that $$\lim_{r\rightarrow 0^+} \int_r^1 t^{x-1}d^{-t}dt=\lim_{n\rightarrow \infty,\ n\in \mathbb{N}} \int_{\frac{1}{n}}^1 t^{x-1}e^{-t} dt$$
65.
