Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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Prove that the set of all algebraic numbers is countable.

I'm a student in Korea. If I make a mistake in grammar, please indicate. Recently, I'm studying the book 'Principles of Mathematical Analysis' So, I tried to solve the exercise #2 in chapter 2. 'A ...
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2answers
270 views

Proving $\text{Li}_3\left(-\frac{1}{3}\right)-2 \text{Li}_3\left(\frac{1}{3}\right)= -\frac{\log^33}{6}+\frac{\pi^2}{6}\log 3-\frac{13\zeta(3)}{6}$?

Ramanujan gave the following identities for the Dilogarithm function: $$ \begin{align*} \operatorname{Li}_2\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right) &=\frac{{...
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1answer
199 views

A rearrangement of an absolutely convergent complex series is also absolutely convergent

I just completed the following proof. Is it valid? Let $\sum_{k=1}^{\infty} a_k$ be an arbitrary convergent series that also converges absolutely. Then $\sum_{k=1}^{\infty} a_k \in \mathbb{C}$ and ...
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159 views

the best constant in an inequality?

I learnt how to show the below inequality by C-S inequality: k is from $0$ to $\infty$ If $\sum a_{k}^{2}9^{k}\le 5$ then $\sum |a_{k}|2^{k}\le 3$. next,I tried to show that 3 is the best possible ...
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1answer
47 views

Decimal representation

I need to prove that, given ${a,n}$ integers with $a<10^n$, $\frac a {10^n}$ has two different decimal representations. I know that this is related to the fact that $0,99999... =1$, and I know how ...
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1answer
97 views

Continuity of a piecewise function

Where is the function $$f: \mathbb R\to \mathbb R,\quad f(x)=\begin{cases}x^2 & x\le 0 \\x+1 & x\gt0\end{cases}$$ continuous? Question from my real analysis class. I know that it is ...
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2answers
507 views

Show that $e^{f(x)}$ is Riemann Integrable

Suppose that $f:\mathbb{R} \to \mathbb{R}$ is Riemann Integrable and $f = 0$ for $f \notin [a,b]$. Show that $e^{f(x)}*\chi_{[a,b]}$ is Riemann Integrable. I think this means that: $g(x) = e^{f(x)} ...
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45 views

Evaluating a limit as $x\to -\infty$

I am trying to evaluate $$ \lim_{x \to -\infty} \left(1+ \frac{1}{x}\right)^{x²}. $$ I'd say it tends to 0, 1 or something linked to $e$ but I have no clue how to prove this... I'm getting really ...
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0answers
54 views

Dirichlet series expansion?

When an analytic function $f(x)$ is given, we can easily obtain the coefficient of $x^n$ in a power series expansion of it. I'd like to know if there exists something similar for Dirichlet series. Is ...
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1answer
33 views

Metric on half-open interval s.t. subset is open w.r.t. $d$ iff open w.r.t. Euclidean metric

I wish to find a metric $d$ on the space $X = (0,1]$ such that $(X,d)$ is complete and so that a subset of $X$ is open with respect to $d$ if and only if it is open with respect to the Euclidean ...
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78 views

Showing that $\lim\limits_{n \to\infty} z_n = A$ implies $\lim\limits_{n \to\infty} \frac{1}{n} (z_1 + z_2 + \ldots + z_n) = A$

In what follows let all values be in $\mathbb{C}$. I'm trying to show that if $$\lim z_n = A,$$ that then $$ \lim_{n \to \infty} \frac{1}{n} (z_1 + z_2 + \ldots + z_n) = A. $$ For ease of ...
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1answer
85 views

Closed graph theorem question?

Let $H$ be a Hilbert space. Let $A:\operatorname{dom}A\to H$ has a closed graph, where $\operatorname{dom}A$ is dense in $H$. Let $S\subseteq \operatorname{dom}A$ be dense. Is it true $A_{|S}$ has a ...
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1answer
110 views

$m(E)=0$ then $m(\lbrace x^2 : x\in E\rbrace$?

Let E be a subset of $\mathbb{R}$ with lebesgue measure zero. How can I prove that $\lbrace x^2 : x\in E\rbrace$ also has lebesgue measure zero? Let $\epsilon>0$, I should find a cover of $\...
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Homework: Second derivative of $\langle Ax, x \rangle$

So let $A \in M_{n}$ and define $f: \mathbb{R}^n \to \mathbb{R}$ by $f(x) = \langle Ax, x \rangle $. Find f' and f''. After some work, I found the first derivative to be $f'(x)(v) = \langle Ax, v \...
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1answer
89 views

Metric topology: boundaries proof

I am stuck on what seems like a completely intuitive proof. A is a subset of X. ( + for disjoint union) I need to show, first, that (i) Closure of S = interior of s + boundary of S Then I am ...
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22 views

Help me correct this proof (Metric Space Topology)

Question: Let $X$ be a topological space. Prove if each point in $X$ is open, then each point in $X$ is closed. Proof: Suppose $\{x\}$ is open for $∀ x∈X$. Pick some arbitrary $x=x_1$. Clearly, $...
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2answers
69 views

Prove $F: \mathbb{R}\to\mathbb{R}$ where $F(x) = \int_a^x f(t)\, dt$ ($a<x$) is surjective

Prove $F: \mathbb{R}\to\mathbb{R}$ where $F(x) = \int_a^x f(t)\, dt$ ($a<x$) is surjective. $f$ is continuous and bounded below by $m>0$. Also $a$ belongs to $\mathbb{R}$ (reals).
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1answer
144 views

$f \in Aut (\mathbb{D})$ with two fixed points is the identity

I have been working on this homework problem for my complex analysis class for some time to no avail. The questions asks for me to show that for $f \in Aut (\mathbb{D})$ such that $f(z_1) = z_1$ and $...
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0answers
41 views

Time series analysis ARCH($m$) model

Consider a stationary ARCH($m$) model $a_t=\sigma_t\epsilon_t$, where $\sigma^2=\alpha_0+\alpha_1a^2_{t-1}+\cdots+\alpha_ma^2_{t-m}$. V. Consider an ARCH(2) model. Write down the predictor of $a_{t+2}...
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63 views

How can I justify combining the limit of a limit into a single limit?

I'm trying to prove this formula for the second derivative of a $f:\, \mathbb{R} \to \mathbb{R}$: $$ \begin{align*} f''(a) &= \lim_{h \to 0} \frac{f'(a)-f'(a-h)}{h} \\ &=...
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94 views

Show that $\sum_{n=1}^{\infty}z^{n!}$ diverges for infinitely many $z$ with $|z|=1$

Problem: I need to show that the power series $\sum_{n=1}^{\infty}z^{n!}$ diverges for infinitely many $z$ with $|z|=1$. I tried to prove it by contradiction by assuming that diverges for finitely ...
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1answer
255 views

Show f(x) = sin(1/x), for x does not equal 0, is differentiable for nonzero real numbers.

Show f(x) = sin(1/x), for x does not equal 0, is differentiable for nonzero real numbers. I was wondering if this would be enough to show the the previous statement: Let c<0 => c does not equal 0....
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1answer
53 views

Find critical points of sin(x*y)

so I got this homework problem that I was having trouble with. The problem is: Let $f(x, y) = \sin(xy)$ defined on all of $\mathbb{R}^2$. Find the critical points of $f$ and classify them as local ...
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1answer
35 views

Prove central symmetry in (0.5;0.5) if f(1-x)=1-f(x)

I would like to prove that if $f:\left[0,1\right]\to\left[0,1\right]$ such that and $f\left(1-x\right)=1-f\left(x\right)$, then $f$ has a central symmetry at $\left(0.5,0.5\right)$. This is ...
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0answers
219 views

Proving differentiability

I'm trying to do Spivak's Calculus on Manifold excersise 2-4. It goes as follows: Let $g$ be a continuous real valued function on the unit circle $\{x\in\mathbb{R}^2:||x||=1\}$ such that $g(0,1)=g(...
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4answers
84 views

Convergence of an infinite sum

Is it possible to use the comparison test for convergence in the following series? $$\sum_{n=1}^\infty \sin \frac 1 n$$ The exercise says that I should find a linear function $f(x)$ that satisfies $...
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1answer
209 views

Is there a ratio (root) test for complex valued sequences?

I am reading a book about complex analysis (Complex Analysis by Ian Stewart/David Tall). Can we use the ratio test (or the n-th root test) for complex valued sequences as well? If so, how can this ...
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3answers
59 views

function has partial derivatives but is not differentiable

Can you write me an example of function which has partial derivatives but is not differentiable? How could I create and prove the function like that?
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95 views

Infinite multidimensional limits

I cannot find the definition of limit of a function of several variables involving infinity such as $(x,y) \rightarrow (0,-\infty)$
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4answers
458 views

How to prove that $\,\,f\equiv 0,$ without using Weierstrass theorem?

Let $\,f:[0,1] \to \mathbb{R}$ continuous. Show that: If $$\int_0 ^1 x^k f(x)\, dx=0,$$ for all $k\in\mathbb N$, then $f\equiv 0$. I know that it can be proved using Weierstrass Theorem, ...
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1answer
59 views

Domains of Lipschitz class are domains of type A.

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. We say that $\Omega$ is of type $A$ if there exists a constant, $A$, such that \begin{equation} |\Omega\cap B_{\rho}(x_0)|\geq A\rho^n \end{...
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1answer
354 views

Can the rational numbers be specified as an ordered field with <order property>?

In other words, (the opposite of my question is) does there exist an ordered field which is isomorphic as (as an ordered SET) to $\mathbb{Q}$? If not, does there exist an order property which ...
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1answer
46 views

Sets A and C with m-1 Elements

If A is a set with m elements and C is a set with one element, then A-C is a set with m-1 elements. What is a proof for this statement?
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1answer
49 views

Set Difference Probability [duplicate]

Here is the question: Prove that for every $\epsilon>0$ and every set $A\in\mathcal{B}(\mathbb{R}^{n})$ there is a compact set $K\subset A$ such that $P(A\setminus K)\leq\epsilon$. --I have ...
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1answer
281 views

An application of Cauchy-Schwarz ineq. on infinite series

If $\sum a_{k}^{2}9^{k}\le 5$ then $\sum |a_{k}|2^{k}\le 3$. sums are from $0$ to $\infty$. could you please help with this question.
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2answers
28 views

Proof Regarding Infinite Sets

If A is an infinite set and B is a finite set prove that A-B is an infinite set.
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1answer
167 views

The Baire space $\mathscr{N}$ is separable

Given is the Baire space $\mathscr{N}$. The elements are functions (or sequences) $f : \mathbb{N} \to \mathbb{N}$ and the metric $d$ is given by $d(f, g) = \frac{1}{k}$ if $f(i) = g(i)$ for all $1 \...
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1answer
102 views

Behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$

I want to analyze the behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$, i.e I want to show it behaves like $e^x(\frac{2}{x}+\frac{2}{3x^2}+...)$ as $x\rightarrow \infty$ I started by ...
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139 views

How can I check the convergence of the sequence? Does it diverge?

How can I check the convergence of the sequence $\frac{1}{\sqrt{n^2+1}}+\frac{2}{\sqrt{n^2+2}}+\cdots+\frac{n}{\sqrt{n^2+n}}$? I think that it diverges,because it is bounded below from $\frac{n(n+1)}{...
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1answer
47 views

Let $f_n \in C([0,2014]) $ Show, that if $f_n \rightrightarrows f$ and for all n $\int_{[0,2014]} ff_n dl_1=0$ then $ f\equiv 0$

Let $f_n \in C([0,2014]) $ Show, that if $f_n \rightrightarrows f$ and for all n $\int_{[0,2014]} ff_n dl_1=0$ then $ f\equiv 0$ I have no idea how to start the exercise like that.
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1answer
313 views

Cardinality of subsets of real numbers

Let A $\subset \mathbb{R}$ be a countable subset of $\mathbb{R}$. Prove that $\mathbb{R}$ and $\mathbb{R}$ \ A have the same cardinality. Use We proved that if X is in finite and B = {x1} $\subseteq$...
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3answers
38 views

can I find the limit in this way?

I have to check if the sequence $b_{n}=\frac{n+cos(n^2)}{n+sin(n)}$ converges.I thought that I could find it like that: $$-1 \leq sin(n) \leq 1 \Rightarrow n-1 \leq n+sin(n) \leq 1+n \Rightarrow \frac{...
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2answers
53 views

Proving an integral using a series

If $f:(0,1]\rightarrow \textbf {R}$ is defined by $f(x)=2nx$ for $\frac{1}{n+1}\leq x \leq \frac 1n$ and $n$ is a natural number, assuming that $\sum_{k=1}^{k=\infty}1/k^2=\pi^2/6$, show that $\int_0^...
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2answers
251 views

If a Laplacian eigenfunction is zero in an open set, is it identically zero?

This seems like it should be easy but I can't seem to figure it out. Suppose $f$ satisfies $\Delta f + \lambda f = 0$ in $\mathbb{R}^n$ (where $\lambda > 0$ is a constant), and in some open set $U$...
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1answer
99 views

Show that if $\sum_{n=1}^{\infty}a_{n}$ converges, $\lim_{n \to \infty}na_{n}=0$.

It is given that $a_{n}$ is a positive and decreasing sequence. Show that if $\sum_{n=1}^{\infty}a_{n}$ converges, $\lim_{n \to \infty}na_{n}=0$. That's what I tried.Could you tell me if it is right?? ...
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1answer
70 views

How to ensure extreme? — using Extreme Value Theorem

I think it's a simple question. How can I ensure the existence of an extreme (maximum or minimum) using the Extreme Value Theorem / Weierstrass theorem? For example, this multivariate case: $$ F(x,y)...
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1answer
77 views

Is the function $f(x)=[x]\sin(πx)$ continuous?

I have a question..Is the function $f(x)=[x]\sin(πx)$ continuous? How can I check it??
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1answer
41 views

Borel measures on $\mathbb{R}$ questions

I am reading a textbook and need some help. First it mentions that we can find a Borel measure such that $\int_\mathbb{R} x^2 \mu(x)<\infty$ but $\int_\mathbb{R} x \mu(x)=\infty$. This seems ...
4
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1answer
795 views

If $f$ is twice differentiable and $f(2^{-n}) = 0 $, for all $n \in \mathbb N$, then $f^\prime(0) = f^{\prime\prime}(0) = 0$.

Let $f : \mathbb R \to \mathbb R$ be a twice differentiable function, such that $f(2^{-n}) = 0$, for all $n \in \mathbb N$ . Show that $$f^\prime(0) = f^{\prime\prime}(0) = 0.$$ My attempt. First, ...
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0answers
93 views

Is it possible to switch limit from inside to outside of integral in this case?

Let $C$ be an open connected subset of $\mathbb{C}$. Let $f:[a,b]\times C \rightarrow \mathbb{C}$ be a function. Assume $f(-,z):[a,b]\rightarrow \mathbb{C}$ is continuous and $f(t,-):C\rightarrow \...