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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22 views

Determining the sign of a function containing ratio

Problem We want to know for which values of $x,y,z,w$ the function $\sigma$ is positive or negative: \begin{equation} \sigma = \frac{A}{A^2-B^2}, \end{equation} where \begin{eqnarray} A & = ...
2
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6answers
209 views

limit laws:$\lim_{n\to\infty}\max(a_n,b_n)=\max(\lim_{n\to\infty}a_n,\lim_{n\to\infty}b_n)$

Let $(a_n)^{\infty}_{n=m}$ and $(b_n)^{\infty}_{n=m}$ be convergent sequences of real numbers. Let $x$ and $y$ be the real numbers $x:=\lim\limits_{n\to\infty}a_n$ and ...
2
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3answers
370 views

Proof that every closed subset of $\mathbb R$ is finite or countable or continuum.

I want to prove that every closed subset of $\mathbb R$ is finite or countable or continuum. I know that for arbitrary subset we can not make similar statements - because of continuum hypothesis. ...
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2answers
152 views

Is $\mathbb{R}\setminus\mathbb{Q}$ a union of countable family of closed sets?

Can we represent set of irrational numbers as union of countable family of closed sets?
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2answers
20 views

Solution of $y'=xy^{1/3}, y(0)=0$ equal to $0$ in $[-c,c]$ and positive for $|x|>c$.

I'm looking for a continuous function $y(x)$ which satisfies the above and trying to make it depend on $c$ so that a solution exists for any $c>0$. I read it is possible, but I can't do it... Can ...
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0answers
27 views

Removable singularities for Dirichlet problems of Laplace equaions?

It is already known that if $u$ is harmonic in $\Omega\backslash\{x_0\}$ where $\Omega$ is a pre-compact domain in $\mathbb{R}^n$, $n\geq2$ and $u=o(|x-x_0|^{2-n})$ when $x\to x_0$, then the singular ...
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1answer
116 views

Verification of Proof that if f(x) is continuous and periodic then it is uniformly continuous on the reals.

Suppose f is defined on all reals. Then there is a positive p s.t. f(x+p)=f(x) for all x. This is my proof: Assume f is continuous on [0,p] then it is uniformly continuous on [-p,p]. Then for x,y ...
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1answer
42 views

Show $(a_n)_{n=m}^{\infty}$ converges to $c$ iff $(a_n)_{n=m'}^{\infty}$ does.

Is my proof correct? Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers. Let $c$ be real number. and let $m' \geq m$ be an integer. Show $(a_n)_{n=m}^{\infty}$ converges to $c$ iff ...
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5answers
85 views

Proof the formula

I have to prove the following formula: $$\sum_{k=0}^n \frac{(-1)^k}{k+1} \binom{n}{k} = \frac{1}{n+1}$$ I do have absolutely no clue ye about how to even start. I'm thinking about using binomial ...
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1answer
51 views

Find k>0 and m∈ℕ so that…

Find $k>0$ and $m∈ℕ$ so that $n^3-7n\ge\ kn$ for all integers $n\ge\ m$. So I am not sure if there is some method I need to follow or if it is sufficient to just pick a random k and m that ...
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2answers
37 views

Is this an inequality law? with division

$a < b$ $c < d$ such that $a \ne b \ne c \ne d$ Will it be true that, $\frac{a}{c} < \frac{b}{d}$ For all positive $a, b, c, d$ Thanks!
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1answer
62 views

an example of a sequence $(u_n)_n$ taking its values in $[-1,+1]$ such that $(u_{n+1}-u_n)$ converge to zero but $(u_n)_n$ does not converge

Define a sequence $(u_n)_n$ by: $$u_n=\cos(\log n).$$ Then, it is easy to show that $(u_{n+1}-u_n)$ goes to zero at infinity. The question is how to prove that $(u_n)_n$ is a divergent sequence ...
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1answer
44 views

Proof of series with induction

I have the sum ...
4
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3answers
126 views

How find this value$\sum\limits_{n=0}^{\infty}\frac{(2n)!}{(n!)^22^{3n+1}}$ [closed]

show that: $$\sum_{n=0}^{\infty}\dfrac{(2n)!}{(n!)^22^{3n+1}}=\left(\frac{1}{2}\right)^{1/2}?$$ this sum is from other problem,if I solve this,then the other problem is solve it
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2answers
39 views

Solving a simple integral by derivating w.r.t. to constants

In the following notes on the solution of the Wave equation by Separation of Variables, in Example 2 the following derivation is given \begin{align*} \int_0^1 x \sin(k\pi x) d x & = \int_0^1 ...
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1answer
50 views

Limit proof for $1/x$ (as $x \to 1$)

Prove $\lim_{x\to 1} \frac{1}{x} = 1$ Using $\epsilon-\delta$ $|\frac{1}{x} - 1| < \epsilon$ for some $|x - 1| < \delta$ $|\frac{1}{x} - 1| = \frac{|1-x|}{|x|}$ Lets require $|x - 1| < ...
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1answer
63 views

Proving that $S=\bigcup_{j=0}^{2^k-1} S_{n-1+k}$ is a spanning set for the $2$-D Baker map

A set $S \subset X$ is a $(n,\epsilon)$-spanning set if $\forall x \in X$, $\exists y \in S $ such that $d_n(x,y)<\epsilon$. This is where we define $d_n(x,y)$ by $d_n(x,y)=\max_{0\leq k < ...
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1answer
96 views

integral of a function

I wanted to find the integral of the function $f(x)$ from zero to one: $$f(x)=\begin{cases}2x\sin(1/x)-\cos(1/x) & : x\in(0,1]\\ 0 & :x=0\end{cases}$$ but I think whether its integral is not ...
5
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4answers
214 views

How prove $\pi^2>2^\pi$

show that $$\pi^2>2^\pi$$ I use computer found $$\pi^2-2^\pi\approx 1.044\cdots,$$ can see this I know $$\Longleftrightarrow \dfrac{\ln{\pi}}{\pi}>\dfrac{\ln{2}}{2}$$ so let ...
2
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1answer
128 views

Function that is uniformly continuous but not bounded?

I've been given the following question but I'm unsure if there are actually any answers: Give examples of functions $f,g: \mathbb{R}\to\mathbb{R}$ which are uniformly continuous such that $f$ is ...
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2answers
57 views

Prove this limit of $x^4 + 1/x$ formally

prove: $lim_{x\to 1} \space \space \space x^4 + \frac{1}{x} $ So, $lim_{x\to 1} \space \space \space x^4 + \frac{1}{x} = lim_{x\to 1} \space \space \space x^4 + lim_{x\to 1} \space \space \space ...
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1answer
26 views

write y(x) according to sin(d+(ay+b)/cx)=y

I end up with the formula $\sin (d+\frac{(ay+b)}{cx})=y $, and try to write $y$ as a function of $x$. There can be multiple solutions (of $y)$ to $\sin(d+\frac{(ay+b)}{cx})=y$ pretending $x$ is known, ...
1
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0answers
41 views

How to calculate these limit superiors and limit inferiors?

Let $\{a_n\}$ and $\{b_n\}$ be two sequences of positive real numbers, such that $$a_{2k} = \frac{1}{3^k}, \, \, \, a_{2k-1} = \frac{1}{2^k},$$ and $$b_{2k} = \frac{1}{4^{k-1}} , \, \, \, ...
6
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2answers
195 views

Inverse of an infinitely large matrix?

This is probably a trivial problem for some people, but I've spent quite some time on it: What is the inverse of the infinite matrix $$ \left[\begin{matrix} 0^0 & 0^1 & 0^2 & 0^3 & ...
5
votes
3answers
97 views

If $\lim_{x \to \infty}f(x)$ exist then $\lim_{x \to \infty}f′(x)=0$

Let $f: [0, \infty) \longrightarrow \mathbb{R}$, where $\lim_{x \to \infty}f(x)$ exist, show that $\lim_{x \to \infty}f′(x)=0$ This fact is clearly intuitive to me but I could not write a ...
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2answers
308 views

A Counter Example about Closed Graph Theorem

This is an example I read around closed graph theorem. Let $Y=C[0, 1]$ and $X$ be its subset $C^\infty[0, 1]$. Equip both with uniform norm. Define $D: X \to Y$ by $f \mapsto f'$. Suppose $(f_n, f_n') ...
1
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1answer
45 views

How can I prove that $\sup(\bigcup_{i \in I} A_{i}) =\sup\{\sup \,A_{i} : i \in I \}$? [closed]

Let $I$ be non-empty set (of "indexes") and for all $i \in I$ let $A_{i} \subset \mathbb{R}$ be non-empty and upper bounded set. How can I prove that $$ \sup \left( \bigcup_{i \in I} A_{i} \right) = ...
2
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3answers
82 views

Show complex solutions exist

Let A be a complex number and B a real number. Show that the equation $\,\lvert z^2\rvert+ \mathrm{Re}\, (Az) + B = 0\,$ has a solution iff $\,\lvert A^2\rvert \geq 4B$. If this is so, show that the ...
5
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1answer
167 views

Solve $x^5 + x - 1 = 0$

Solve $x^5 +x - 1 = 0$ I am simply curios to see how the solution would go, since it is a quintic, it cannot be done by regular methods. Im just curios to see what people come up with (I can't solve ...
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3answers
66 views

Sequence which converges pointwise but not uniformly?

it might be simple but I don't find a sequence $f_n: [0,1] \rightarrow \mathbb{R}, n \in \mathbb{N}$ that converges pointwise but not uniformly. First I thought it could be $f_n(x) = \frac{x}{n}$ but ...
0
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1answer
73 views

If a sequence ${a_n}$ is monotonically increasing. then $\lim_{n \to \infty} a_n = \sup{(a_n)}$

Can you please tell me if my proof is correct: If a sequence ${a_n}$ is monotonically increasing. Then $$\lim_{n \to \infty} a_n = \sup{(a_n)}$$ Proof: $$a_n\leq a_{n+1}\leq \sup(a_n)$$ Assume ...
1
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1answer
31 views

Why is it that the derivative of $f(y(x))$ with respect to $x$ vanishes?

I have been messing around with the Euler-Lagrange-Equation lately and found that one could use the Beltrami Identity with a function like this: $$f(y,y') = \sqrt{\frac{1+y'^2}{-2gy}}~, ~~where~ y = ...
2
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1answer
66 views

How to use contour integration to compute a **real** integral?

Suppose we are given: $$\text{Evaluate} \int_{0}^{1} \frac{1}{1+x^2} \text{dx}$$ This is quite easy because you will notice that: $$\int_{0}^{1} \frac{1}{1+x^2} \text{dx} = \arctan(1) - \arctan(0) ...
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2answers
216 views

For any positive real number $x>0$, there exist a positive integer N such that $x>1/N>0$.

I am trying to prove the following: For any positive real number $x>0$, there exist a positive integer N such that $x>1/N>0$. What I know is: since x is a real number, it can be expressed ...
3
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1answer
64 views

A density question

Let $\theta \in \mathbb{R} \setminus \mathbb{Q}$. Is the set $\{ (2n+1) \theta \bmod 1: n \in \mathbb{N} \}$ dense in $[0,1]$?
3
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1answer
87 views

spring representation of graphs

Suppose we have a finite graph $G$ which we want to embed in ${\bf R}^d$; fix the positions of some nodes and connect all the nodes of the graphs with ideal springs of varying strength; (i.e. there is ...
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1answer
22 views

Show that: $\mid\det(A)\mid\mu^*(M)=\mu^*(A(M)) $.

Let $n\in\mathbb{N},M\subset\mathbb{R}^n$ with $\mu^*(M)$ being finite and $A$ a linear mapping, which is diagonal to the standard basis of $\mathbb{R}^n$. Show that: ...
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1answer
147 views

Measurable functional calculus

I am struggeling with this exercise: Let $T \in L(H)$ be a self-adjoint operator and $\Psi$ be a measurable (Borel) functional calculus on the spectrum of $T$. For a Borel set $\Delta \subset \sigma ...
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3answers
89 views

Limit of a sequence $\sqrt[n]{3} $

Using a definition of limit of a sequence and Bernoulli's inequality proof that limit of $\sqrt[n]{3} $ is 1. From the definition I know that ∀ε>0 ∃N∈ℕ ∀n>N |an-g|<ε and g=1. ...
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1answer
117 views

Lp spaces are nested but then why is 1/x square summable but not summable?

If $1\leq s<r<\infty$ and $f\in L^r$ then $f\in L^s$, so then why is $\frac{1}{x}$ not in $L^1$ but is in $L^2$ for the counting measure $c:\mathbb{N}\rightarrow \mathbb{R}$?
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1answer
40 views

Limit of a summation, using integrals method

I have seen an interesting question on stackexchange, which I would like to requote so that I can understand the answer =) $\lim\limits_{n\to\infty} \dfrac{1^{99} + 2^{99} + \cdots + ...
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2answers
117 views

Is $\ell^1$ an inner product space?

Considering the parallel result for $L^p$ spaces, I would guess that $\ell^1$ is not an inner product space. The proof would presumably follow by providing counterexample sequences $x = (x_n) $ and ...
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2answers
26 views

Converges of a sequences defined through a continued fraction

Consider the following sequence $(b_n)_{n \geq 1}$ recursively given through the continued fraction $b_1 = \frac{1}{1}$, $b_2 = \frac{1}{1+ \frac{1}{2}}$, $ \dots , b_n = \frac{1}{1+b_{n-1}}$ ...
0
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2answers
953 views

Sequences of polynomial functions converging uniformly on $[a,b]$ to a continuous function not a polynomial

What is (are) the necessary and sufficient condition(s), if any, for a sequence of polynomial functions to converge uniformly on a given (finite) closed interval $[a,b]$ to a continuous function not a ...
0
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2answers
227 views

Prove that the sequence $(f(x_n))_{n\geqslant1}$ is Cauchy.

Let $f:[0,2]\to\mathbb{R}$ be a regulated function. Let $(x_n)_{n\geqslant1}$ be a sequence in $[0,1)$ with $\lim_{n\to \infty}x_n=1$. Prove that the sequence $(f(x_n))_{n\geqslant1}$ is Cauchy. I ...
0
votes
3answers
85 views

How to evaluate this $1/n$ infinite sum?

How to evaluate$$\sum ^{\infty}_{n=1} {e}^{-n}$$ without using the easy-formula. We easily notice a pattern. $$\begin{align} S_1 &= e^{-1} \\ S_2 &= e^{-2} + e^{-1} = \frac{1 + e}{e^2} \\ ...
1
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1answer
234 views

Prove that f(x) is regulated.

Define $f:[0,1]\to \mathbb{R}$, $f(x):=0$ if $x\notin \mathbb{Q}$, $f(p/q):=1/q$, $q>0$, $p, q$ coprime integers. Prove that $f$ is regulated. A function $f:[a,b]\to\Bbb R$ is a regulated ...
0
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1answer
47 views

solving equations with log and polynomials.

I need to solve/estimate for x in the following equation - $Klnx + x^\beta = r$. $K,r > 0.$ An estimate for large r(fixed K) is what I am looking for.
3
votes
1answer
43 views

How prove this Ratio Test and Its Generalizations problem?

Question: let $\alpha\in (0,1)$,and the postive sequence $\{a_{n}\}$ such $$\lim_{n\to\infty}\inf \left(n^{\alpha}\left(\dfrac{a_{n}}{a_{n+1}}-1\right)\right) =\lambda\in (0,+\infty)$$ show ...
4
votes
1answer
396 views

How is the interchange of the limit and the maximum valid at this point in Erwin Kreyszig?

In 1.5-5 in Erwin Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS, the author shows completeness of the space $C[a,b]$ of all (real- or complex-valued) functions defined and continuous ...