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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Advanced Calculus Question. Prove (sn + tn) is a Cauchy sequence

Based on the definition of a Cauchy sequence, that if (sn) is a Cauchy sequence and (tn) is a Cauchy sequence, then (sn + tn) is a Cauchy sequence I try to work from the definition that |(Sn + tn) ...
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0answers
14 views

Regular surface/normal line

Let $\mathcal A$ be a regular surface in $\mathbb R^3$ and $P$ a point in $\mathbb R^3\setminus\mathcal A$. Suppose that $C$ is a point at minimum distance from $P$. Show that $P$ belongs to the ...
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0answers
13 views

computational question concerning singular integral theory

(I have posted this question yesterday, but it remained unanswered. I am changing the title of the question and posting it again hoping that other might pay attention) Let $m\in ...
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0answers
8 views

Can anybody help-me whit my Partial differencial equations?

Let $a_i, i=1,\cdots,n,$ $\cal{C}^1$ functions over $\mathbb{R}^n\times[0,\infty)$ such that $$\mid a\mid\leqslant\dfrac{1}{k}, \text{where}\; a=(a_1,\cdots,a_n).$$ In adtion, we have ...
2
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3answers
63 views

Possible new definition of Gamma (Euler-Mascheroni Constant): $\lim_{x \to 0} (-\ln ( \sqrt[x]{x!} )) = \gamma$

I think I've discovered a new definition for the Euler-Mascheroni Constant (Gamma) I can't find it online anywhere, has anyone seen it before? $$\lim_{x \to 0} (-\ln ( \sqrt[x]{x!} )) = \gamma$$
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1answer
21 views

Continuous functions unbounded on set

For Each of the sets construct a continuous function that is unbounded on the set. $\Bbb N$ $(2,3)$ $\left\{\frac 1 n \mid n \in \Bbb N\right\}$ $[0, \sqrt 2]\cap \Bbb Q$ ...
2
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1answer
18 views

Basic analysis question: $\max_{1\leq i\leq n} a_i\geq n\epsilon\implies a_n\geq n\epsilon$

This is a follow up to something I asked earlier. (This question is self-contained so you don't need to click the link.) Thank you very much for your help! Question: $\{a_n\}$ is a sequence of ...
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1answer
12 views

Analysis: Proof checking and help on 2nd part (Integrals)

So I have the question, $f(x)= x$ if $0$ $\leq$ $x$ $\leq$ $1$ and $f(x)= x+2$ if $1<x$ $\leq$ $2$ (the same f(x) I just couldn't figure out how to do the big bracket) Part 1 is asking me to ...
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1answer
17 views

Show that $\int_{X}u\, \mathrm{d}\mu\leq 4$ and $\int_{X}u\, \mathrm{d}\mu=1$.

Let $(X,\mathcal{A},\mu)$ be a measureable space. Let $u\in \mathcal{M}_{\mathbb{R}}^{+}(\mathcal{A})$ and $\lbrace u_{j}\rbrace_{j\geq 1}$ be a sequence of functions in ...
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4answers
33 views

Verifying my proof that if $|S(x)| \leq 1$, then $\lim_{x \to 0} x\cdot S(1/x) = 0$

Question: Suppose that $S : R \to R$ is a function so that for all $x$, $−1 \leq S(x) \leq 1$. Prove from the limit definition that $$\lim_{x\to0} x \cdot S(1/x) = 0.$$ This is my ...
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0answers
22 views

Find the pointwise limit of {gn} on [0, ∞). Please help!!

Consider sequence of functions gn(x)=x\over 1+x^n over [0,\infty) (a) Find the pointwise limit of {gn} on [0, ∞). g(x)= x 0 \le x \lt x 1/2 x=1 00 x>1 Show gn(x) ...
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0answers
8 views

First order PDE with discontinuous coefficients

I want to consider the following equation $$u_t+\mathrm{sgn}(x)u_x=0,\,\,u(0,x)=u_0(x)$$ Now if $x>0$ or $x<0$ I can use the method of characteristics to obtain $u(t,x)=u_0(x-t)$ if $x>t$ and ...
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0answers
22 views

Strange inequality

I found the inequality $\beta e - \frac{3}{2} n \ log(e+Bn)+ \frac{5}{2} \ n \ log(n) + const \cdot n \geq \frac{\beta e}{2}+ \beta n $ in a textbook,provided that either $e$ or $n$ is large. We ...
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0answers
24 views

Prove that this function is Borel measurable

Prove that if $s\ge 0$, $f:\mathbb{R}^n\to\mathbb{R}^m$ is continuous and $K\subset\mathbb{R}^n$ is compact, then the function $$ F:\mathbb{R}^m\to [0,\infty]\\y\mapsto H^{s}(K\cap f^{-1}(\{y\})) $$ ...
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0answers
33 views

How to show $\exp(tX)\exp(tY)=\exp(t(X+Y)+tR(t))$ with $\displaystyle \lim_{t\to 0} R(t)=0$?

Let $X\in GL(n, \mathbb R)$. The exponential of $X$ is the matrix given by $$\exp(X)=\sum_{n=0}^\infty \frac{X^n}{n!}.$$ I need some help for showing the following result: ...
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0answers
22 views

Determine integrals $\int_{\mathbb{R}}u\, \mathrm{d}\delta_{3}$ and $\int_{\mathbb{R}}u\, \mathrm{d}\delta_{\pi}$.

Consider the function $u:\mathbb{R}\to [0,\infty]$ given by $$ u(x)=\sum_{n=1}^{\infty}\frac{1}{n^{2}}1_{[n,n+1]}(x) $$ I have determined that $\int_{\mathbb{R}}u\, \mathbb{d}\lambda=\pi^{2}/6$ where ...
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1answer
38 views

Ode with step function in the right-hand-side

I want to solve the following ODE: $$\dot{X}(t,x)=F(X(t,x))$$ where $F(x)=1$ if $x>0$ and $-1$ if $x<0$. How to treat this discontinuous right-hand-side?
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2answers
25 views

Limits of functions in metric spaces

My teacher said that in the definition of limit, the point in the domain, must be of accumulation, because otherwise the limit is not unique. Why? If the point is isolated, the function is continuous, ...
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1answer
57 views

Prove (or disprove) this $\sum_{n=1}^{\infty}\dfrac{a_{n}}{n}$ is convergence? [on hold]

Let $a_{n}>0$ be a sequence, and $0<a\le 1$, such that $\sum\limits_{n=1}^{\infty}(a_{n})^a$ converges. Prove or disprove $\sum\limits_{n=1}^{\infty}\dfrac{a_{n}}{n}$ is convergent. I ...
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0answers
49 views

Acceptance of Facts in Mathematics [on hold]

I have a simple question about acceptance of conventions/facts in mathematics. Tell me: Is it an accepted fact in the world of mathematics that something written like $\sin(x)$ would be considered ...
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1answer
23 views

Time derivative of operator

I have to compute, at least formally, the following derivative $$\partial_t \exp(it\Delta)f(x-ct)$$ where $\Delta$ is the Laplacian and $c$ is a constant. I know that $e^{it\Delta}$ is the Schrodinger ...
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0answers
14 views

Theorem about n=1 wave equation in Evans

In Evans, PDE edition 2 on p68 we have a Theorem that tells us some properties about the solution to the wave equation for $n=1$. It reads: Assume $g \in C^2(\mathbb{R})$, $h\in C^{1}(\mathbb{R})$, ...
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1answer
20 views

Let A be a bounded infinite subset of R^2, show that A has at least a limit point.

Let A be a bounded subset of R^2 with infinite points, show that A has at least one limit point. How can a prove that without using compactness?
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1answer
14 views

Correction of Proof that if f:[0,1] is a continuous function and f(x)>2 with x being in [0,1) it is not necessary that f(1)>2

Here's how I wrote it up: To approach this consider an example of a continuous function which fails to satisfy f(1)>2 even though it satisfies f(x)>2 for x in [0,1). My counterexample is a negative ...
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0answers
20 views

Show that there is a θ, depending on α such that 0 < θ < 1

Here is the question that I am struggling, please help.
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0answers
29 views

Method of characteristic for second order pde

Can I use the method of characteristic to solve second order pdes? For instance I canconsider the equation $$u_t+u_x=u_{xx}$$
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1answer
29 views

Sequence of functions that converges a.e. but not in the $L^1$ norm

How can I construct a sequence of functions that converges a.e. but it does not in the $L^1$ norm?
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2answers
33 views

$f(x)=1$, for every $x \in [0,1]$ if $f:[0,1]\to\mathbb R$ is continuous and $f(p)=1$ for every $p\in [0,1]\cap\mathbb Q$.

How would you approach this if I have to use the fact that "every number is a sequence of rational numbers"? Currently, I am proving this by contradiction in the following way: Let f(p)=1 for all ...
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0answers
30 views

Given two sets $A,\; B$ and that $|A| = |B|$, show that $|2^A| = |2^B|$

Given two sets $A,\; B$ and that $|A| = |B|$, show that $|2^A| = |2^B|$. Intuitively, I think this is true, but I am having trouble showing this formally. I know that there exists a bijection $f: A ...
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2answers
51 views

If $\sum a_n<\infty$ then $\exists (b_n)$ [duplicate]

If $\displaystyle\sum_{n=1}^\infty a_n<\infty$ and $a_n>0$, then $\exists (b_n)$ such that $b_n\geq1$, $b_n\to+\infty$ and $\displaystyle\sum_{n=1}^\infty a_nb_n<\infty$.
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0answers
14 views

P-a.s inequality meaning

If I have the following definition [ \begin{split} \mathcal{L}^0(\mathbb{R}^N)&=\mathcal{L}^0(\Omega,\mathcal{F},\mathbb{P},\mathbb{R}^N):=\{\textbf{X}=(X^1,\cdots,X^N)|X^n\in ...
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0answers
8 views

Proving properties of nth roots

First let me define some things. Let $x \gt 0$ and $n \ge 1$. Now $x^{\frac{1}{n}}:=\sup\ [ y\in \mathbb R : y \ge 0 \text{ and } y^n \le x]$ (a) If $x \gt 1$ then $x^{\frac{1}{k}}$ is a decreasing ...
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0answers
9 views

characterisation of continuity of a function in two variables in polarcoordinates

As the the titel of my question already indicates the question I have is about continuity. I know the "$\epsilon-\delta$ Definition" of continuity and think I have understood it. On the internet I ...
3
votes
2answers
72 views

Convergence of $\sum \frac{(2n)!}{n!n!}\frac{1}{4^n}$

Does the series $$\sum \frac{(2n)!}{n!n!}\frac{1}{4^n}$$ converges? My attempt: Since the ratio test is inconclusive, my idea is to use the Stirling Approximation for n! $$\frac{(2n)!}{n!n!4^n} ...
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0answers
20 views

Integrating a particular function.

Could someone please show me how to integrate the following: $\int_{-\infty}^{\infty}p(x)L(p(x))dx$, where $L(x)$ is defined as $L(x)=\frac{x-1}{ln(x)}$ and $p(x)=(2\pi ...
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1answer
34 views

Show set is countable.

I have a set $A = \left\lbrace n^2+m^2:n,n \in \mathbb{N}\right\rbrace$. I need to show that this set is countable, and I don't know if im using the right method. So far i got this: $\mathbb{N}$ is ...
1
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1answer
65 views

How prove this integral equation has a unique solution

Question: Consider the equation $$3u(x)=x+(u(x))^2+\int_{0}^{1}|x-u(y)|^{1/2}dy$$ show that it has a continous solution $u$ satisfying $0\le u(x)\le 1$ for $0\le x\le 1$ this problem From page ...
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0answers
21 views

Compactness of the convergent to zero sequences

I've gotta prove that $$T = \left\{ \left\{ x_i \right\} \in {\ell ^\infty }:\left| x_i \right| < \mu_i,\mathop \lim\limits_{i \to \infty } \mu _i = 0 \right\} \subseteq \ell ^\infty $$ is ...
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1answer
33 views

Am I going about this the right way?

Show that for any $α ∈ R$, there exist infinitely many rational numbers $\frac{m}{n}$ with $|α − \frac{m}{n^2}| < \frac{1}{n}$. So we know that $-1≤\frac{1}{n}≤1$ which implies $\frac{1}{n^2}≤1$. ...
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3answers
50 views

How find this limit $\lim_{x\to 0}\frac{\int_{0}^{x}\sin{t}\ln{(1+t)}dt-\frac{x^3}{3}+\frac{x^4}{8}}{(x-\sin{x})(e^{x^2}-1)}$

Find this limit $$I=\lim_{x\to 0}\dfrac{\int_{0}^{x}\sin{t}\ln{(1+t)}dt-\dfrac{x^3}{3}+\dfrac{x^4}{8}}{(x-\sin{x})(e^{x^2}-1)}$$ I think $$I=6\lim_{x\to ...
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3answers
42 views

Show $|\sin(y)y - \sin(x)x| \leq C|y - x|$ for some $C > 0$

Show $|\sin(y)y - \sin(x)x| \leq C|y - x|$ for some $C > 0$. This is one of the steps in a bigger problem I'm trying to solve, and while it first appeared it would be entirely straightforward, I ...
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1answer
14 views

Annutiy choices question

If an annuity pays 4800 annually with a 2% increase per year or has an option of 6000 annually, how many years will the total amount paid is equal in both options Each year is 4800 + 4800 x 1.02 + ...
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0answers
29 views

Tricky Change Of variables?

If $f(x,y,z)$ is a differentiable function then from $\mathbb{R}^3$ to $\mathbb{R}$ then: if $w_1:=x+y$, $w_2:=\frac{x-y}{x+y}$ and $w_3:=x^2+y^2+z^2$ then what would the function resulting from a ...
1
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1answer
28 views

$k^m$ where $k$ is infimum of $\{x \in \mathbb{Q} | m \le x^m\}$

if $k = \inf\{x \in \mathbb{Q} | m \le x^m\}$ can I say that $k^m = m$? Can you please explain why yes or why not? Any help is appreciated. Thank you
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0answers
8 views

Show that $ f$ is strongly differentiable at $x_0$ .

Definition: Let $U\subseteq \Bbb R^m$ be an open set. Let $f: U \to \Bbb R^n$ be a function and $T: \Bbb R^m \to \Bbb R^n$ be a linear transformation. We say that f is strongly differentiable ...
0
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1answer
24 views

Let $A$ be an infinitely countable set and let $B$ be a finite set. Show that $A \cup B$ is also countable.

Let $A$ be an infinitely countable set an let $B$ be a finite set. Show that $A \cup B$ is also countable. In the solution for this exercise, a function $f: \mathbb{N} \to A \cup \left ( B \setminus ...
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0answers
13 views

Show that if f is strongly differentiable at $x_0$ then it satisfies Lipschitz condition in a neighbourhood of $x_0$.

Definition: Let $U\subseteq \Bbb R^m$ be an open set. Let $f: U \to \Bbb R^n$ be a function and $T: \Bbb R^m \to \Bbb R^n$ be a linear transformation. We say that $f$ is strongly differentiable ...
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2answers
34 views

Show that there exists a subsequence $\{F_{n_{k}}\}$ which converges to uniformly on $[a,b]$.

Let $\{f_n\}$ be uniformly bounded sequence of functions which are Riemann-integrable functions on $[a,b]$ and define for $a\leq x\leq b$. $$ F_n(x)= \int_a^x f_n(t)dt.$$ Show that there exists a ...
0
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0answers
37 views

If f is a continuous function on $[0,1]$ and if for each $n=0,1,2,3,…$. Prove that $f=0$ on $[0,1]$.

If f is a continuous function on $[0,1]$ and if for each $n=0,1,2,3,...$ $$ \int_0^1 f(t)t^n dt=0$$ Then prove that $f=0$ on $[0,1]$. Here I don't want the proof. I have one proof. But I have ...
2
votes
3answers
66 views

How can we show that $ \sum_{n=1}^{\infty} \frac{n}{2^n} = 2 $? [duplicate]

How can we prove the following? $$ \sum_{n=1}^{\infty} \frac{n}{2^n} = 2 $$ It would be great to see multiple ways, or hints, about how this can be proven. I know this is a power series ...