Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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7 views

Prove that the function $ξ\in R \mapsto {e^{i\cdot ξ\cdot λ}-1\over i\cdot ξ}-λ$ is $C^{\infty}$ is $C^{\infty}$

Prove that the function $$ξ\in R \mapsto {e^{i\cdot ξ\cdot λ}-1\over i\cdot ξ}-λ$$ is $$C^{\infty}$$ (and in the point of ξ=o) Any ideas how to prove this? i am trying to think some ideas but i can ...
0
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0answers
5 views

State of a $C^*-$ algebra A.

For a nondegenerate $*-$representation $\pi$ of a $C^*-$ algebra $A$ and an element $h \in H$ with $||h||=1$, define $f_h: A \rightarrow \mathbb{C}$ by $f_h(a)=<\pi(a) h | h>.$ How can we show ...
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0answers
12 views

multiplicative inverse of a medible function

I want to prove that if $f$ and $g$ are lebesgue measurable functions then $h$ defined by: $$ h(x):=\frac{f(x)-g(x)}{f(x)+g(x)} \text{ if } f(x)+g(x)\neq 0 \\ 0\text{ if } f(x)+g(x)= 0 $$ ...
0
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1answer
25 views

if g is not constant zero, $f\circ g$ has a local minimum at zero

Consider $f:\mathbb{R}^2\to\mathbb{R}\; f(x,y)=(x^2-y)(x^2-3y)$ and a linear function $g:\mathbb{R}\to\mathbb{R}^2,\; x\mapsto \begin{pmatrix} g_x(x)\\ g_y(x) \end{pmatrix} $. The claim is: If $g$ ...
0
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1answer
27 views

What does this notation mean? Functional Analysis

I am studying analysis at the moment and came across this notation and I would like to know what it really means: $$C_{c}^{\infty}(\Omega)$$ My understanding so far is that,this is the space of ...
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0answers
28 views

Exercise to Heat equation

I am dealing with the following exercise on a generalized heat equation Let $U$ be a open bounded subset of $\mathbb{R}^n$ and $T \in (0, \infty).$ Now, let $u$ be a solution to $$(\partial_t - ...
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2answers
20 views

Recurrence sequences with two initial condition: how do I calculate the limit?

I've done some exercises with recurrence sequences with one initial condition. So, now that I'm attempting one exercise with two initial conditions I'm confused. Could you show me what to do? Let ...
0
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1answer
38 views

Limit of $a_{n+1}= \frac{n}{n+1} a_n$

I think that this sequence $$a_{n+1}= \frac{n}{n+1} a_n$$ can be rewritten as $$a_n= \frac{1}{n+1}a_0.$$ Therefore the limit should be $0$. But my proof by induction turns out wrong. Is my idea ...
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2answers
27 views

Counter example in Sobolev spaces: Is there a standard example for showing that $H^1(\mathbb{R}^2)\not\subset L^\infty(\mathbb R^2)$?

One way to define the Sobolev space $H^s(\mathbb{R}^n)$ is as the completion of the $C^\infty$ functions on $\mathbb{R}^n$ under the norm $$\lVert f \rVert_{H^s(\mathbb{R}^n)}:= \left( ...
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1answer
33 views

Bring a proof for the fundamental theorem of calculus.

If $f\in \mathscr{R}$ on $[a,b]$ and if there is a differentiable function $F$ on $[a,b]$ such that $F'=f$, then $$\int_a^b f(x)\ \ d(x)=F(b)-F(a)$$
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3answers
41 views

Find the limit of the sequences: $a_{n+1}=3a_n - n + 1$ and $(a_n)^\frac{1}{n}$ with $a_0 > 0 $

Let $a_0 > 0 $ and $$a_{n+1}=3a_n - n + 1.$$ I have to find its limit. I have also to find the limit of $(a_n)^\frac{1}{n}$. But this seems even more complicated. For the first part I've used the ...
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1answer
21 views

Convergence of $u * \eta_\epsilon$

Let $\eta \in C_c^\infty(B(0,1)), \eta \ge 0, \eta$ radially symmetric and $\int_{\mathbb{R}^n} \eta d\mathcal{L}^n = 1$. $\eta_r := r^{-n} \eta(\frac{x}{r}) \in C_c^\infty(B(0,r))$. Integral of ...
2
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3answers
54 views

Prove that a certain sequence is increasing and find its limit: $a_1 = 1$ and $a_{n+1}=n(1+\ln a_n)$ (and $(a_n)^\frac{1}{n}$)

Let $a_1 = 1$ and $$a_{n+1}=n(1+\ln a_n).$$ I have to find its limit. I want to prove that it is increasing for starters, but I'm already stuck. What should I do? I have also to find the limit of ...
0
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1answer
20 views

Little o notation within another little o

To prove $e^{x + o (x)} = 1 + x$ as $x \rightarrow 0$, I can do it directly: $\lim_{x \rightarrow 0} \frac{\log (1 + x) - x}{x} \overset{\text{l'hopital}}{=}\lim_{x \rightarrow 0} \frac{(1 + x)^{- ...
2
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0answers
42 views

A necessary condition to $F'(x)=f(x)$ for a continuous function $f$

Consider , $$F(x)=\int_a^xf(t)\,dt$$ If the function $f:[a,b]\to \mathbb R$ is continuous then $F(x)$ is differentiable and $F'(x)=f(x).$ I know that the continuity condition of $f$ is ...
-2
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0answers
27 views

How can the elements $a_1, a_2, a_3\ldots, a_n$ be distinct in Theorem 2.13 of Rudin?

In Theorem 2.13 of Rudin, how could the elements $a_1,a_2,\ldots, a_n$ be distinct like he says they can? $A$ is a countable set (or just a set) and, therefore, all elements must be distinct. Perhaps ...
2
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1answer
22 views

Radius of convergence of a power series with $a_n$ convergent

Let $\{a_n:n \geq 1\}$ be a sequence of real numbers such that $\sum_{n=1}^{\infty} a_n$ is convergent and $\sum_{n=1}^{\infty} |a_n|$ is divergent. Let $R$ be the radius of convergence of the power ...
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0answers
10 views

If $X=\{0,1\}$, there exists an outer measure $\mu^*$ on $X$ such that $\mu^* \neq \mu^+$

Background Let $\mu^*$ be an outer measure on $X$ , $\mathcal{M}^*$ the $\sigma-$ algebra of all $\mu^*$ measurable sets, $\overline{\mu}=\mu^*\bigg|_{\mathcal{M}^*},$ and $\mu^+$ the outer measure ...
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1answer
23 views

If $f$ is a convex function in $(a,b)$ then $|f|$ is convex function in $(a,b)$.

TRUE or FALSE: If $f$ is a convex function in $(a,b)$ then $|f|$ is convex function in $(a,b)$. My Proof: Since $f$ is convex function so, $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$ , for all ...
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1answer
33 views

Explanation of taylor series

I understand that for a Taylor series of a function $f(x)$, centered around the point a, the general expression can be written as: $$ \begin{align} &f(x) \\ &= f(a) + f'(a) (x-a) + ...
1
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1answer
37 views

Cauchy Sequences To Prove $f(z)$ is not continous [on hold]

I've learn ways to prove the discontinouity of a complex function. I have not learn Cauchy Sequences however. I cannot find useful information on the subject. Please explain
4
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0answers
36 views

Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$ f(x)=\sum_{n\geq 0}a_n x^n $$ converges absolutely for all $x$. Under ...
1
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0answers
15 views

Analytic function with alternating taylor series

Is there a characterization of the sequences of non-negative real numbers $\{a_n\}_{n=0}^{\infty}$ such that $$ f(x)=\sum_{n\geq 0}a_n (-x)^n $$ converges absolutely for all $x\geq 0$? It's not hard ...
2
votes
4answers
93 views

Compute $\lim_{n \to \infty} \left(\frac{1}{\sqrt{n^3+1}} + \frac{1}{\sqrt{n^3+4}} + \cdots + \frac{1}{\sqrt{n^3+n^2}}\right)$

How do I evaluate the following limit? I guess I should do a comparison, but I've got no clue about what to do. Could you give me a hand? $$\lim_{n \to \infty}\left( \frac{1}{\sqrt{n^3+1}} + ...
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1answer
41 views

About summer course or online course of Linear algebra and real anyasis [on hold]

I just looking for the online course for Linear algebra or real analysis but it should be upper level. i saw MIT and another college but our university said it was not upper level its likely ...
0
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2answers
41 views

Calculate: $\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$ and $\lim_{n \to \infty} \frac{10^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$

I have to evaluate the following limits (which are similar). However, I don't know how to evaluate them. Could you give me a hand? $$\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln ...
4
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3answers
104 views

How can I calculate $\lim_{n \to \infty} (1 + \frac{1}{n!})^n$ and $\lim_{n \to \infty} (1 + \frac{1}{n!})^{n^n}$?

How do you calculate the following limits? $$\lim_{n \to \infty} \left(1 + \frac{1}{n!}\right)^n$$ $$\lim_{n \to \infty} \left(1 + \frac{1}{n!}\right)^{n^n}.$$ I really don't have any clue ...
2
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1answer
43 views

Verify solution to ODE

I am given the ODE $$\left(f''(x)+\frac{f'(x)}{x} \right) \left(1+f'(x)^2 \right) = f'(x)^2f''(x)$$ and I already know that the solution to this ODE is given by $$f(x)= c \cdot arcosh \left( ...
-4
votes
2answers
61 views

Why is $n^\sqrt n - 2^n \to - \infty$ and $\sqrt n ^n - 2^n \to +\infty$ [on hold]

Can you explain technically why the following limits are correct? $$n^\sqrt n - 2^n \to - \infty$$ and $$\sqrt n ^n - 2^n \to +\infty$$
-1
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2answers
48 views

How do I calculate the following limit? $\lim_{n \to \infty} n ((8 + \sin (2^\frac{1}{n}))^\frac{1}{3} -2)$

How do I calculate the following limit? I'm short on ideas for this one: $$\lim_{n \to \infty} n ((8 + \sin 2^\frac{1}{n})^\frac{1}{3} -2)$$
0
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0answers
12 views

Relation between Ill-posed problem and eigenvectors?

This question is related to the question below: Is there a relation between Ill-posed problems and Eigenvectors. In the answer of the above question, it was shown that the ill-posed problem can be ...
3
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4answers
74 views

How do I calculate $\lim_{n \to \infty} n^\frac{1}{n} (n+1)^\frac{1}{n+1} … (2n)^\frac{1}{2n}$

How do I calculate the limit of the following sequence? $$\lim_{n \to \infty} n^{\frac{1}{n}} (n+1)^\frac{1}{n+1} ...... (2n)^\frac{1}{2n}$$
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0answers
22 views

At what point in size are Tidal Forces not visible? [on hold]

From my knowledge, if you were to drop a very large ball high from Earth, it would be stretched vertically and compressed horizontally. If this continued to happen to the point that the object split ...
0
votes
1answer
12 views

Find limit inferior and limit superior of $[1+\sin n]$ and $n - [\sqrt n]$

I have to find the limit inferior and limit superior of the following sequences: $$[1+\sin n]$$ and $$n - [\sqrt n].$$ I have done similar exercises before, but never with the integer part function ...
4
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0answers
29 views

Probabilistic interpretation for representation of unity using the zeta function

There's a cute identity, I believe due to Borwein, Bradley and Crandall (Section 4): $$1=\sum_{n=2}^\infty (\zeta(n)-1).$$ There are some generalizations in the linked paper as well. Question: Is ...
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0answers
32 views

An absolutely continous function from $[0,1]$ to $\mathbb{R}$ that is not $C^{1}$

I need to find a function $f:[0,1]\rightarrow \mathbb{R}$ that is absolutely continuous, but not $C^1$. My guess is $f(x)=|x-1/2|$. Is it enough to say that $f'(x)$ exists almost everywhere, and ...
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0answers
32 views

real analysis questions [on hold]

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1answer
22 views

derivative of $f(r\cos\phi,r\sin\phi)=r^a\cos(a\phi)$

Let $$ f(r\cos\phi,r\sin\phi)=r^a\cos(a\phi) $$for some $r\in(0,\sigma)\subset\mathbb R$ and $\phi\in (0,\rho)\subset(0;2\pi]$. How do you calculate $Df=(\partial_1 f,\partial_2 f)$ ? I thought ...
1
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0answers
18 views

Envelope of a family of lines. When does it exist?

Given a family of lines $(\gamma_t): a(t)x+b(t)y+c(t)=0,\ t\in I$ (interval in $\mathbb{R}$), where $ a,b,c:I\to\mathbb{R},\ a^2(t)+b^2(t)\neq 0,\ \forall t\in I$ , what conditions should we impose ...
3
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2answers
83 views

Showing $r\in\mathbb Q\setminus \{0\}\implies e^r\notin \mathbb Q$

For a given $n>0$, let $\displaystyle J_n:x\to \frac{1}{n!}\int_{-x}^x(x^2-t^2)^ne^tdt$ a. Prove that there exists $A_n,B_n\in \mathbb R_n[X]$ such that $\forall x\in \mathbb R^+, ...
0
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5answers
55 views

How to evaluate $\lim\limits_{x\to1}\frac{x^{m+1}-x^{n+1}+x^n-mx+m-1}{(x-1)^2}$?

How to evaluate $$\lim\limits_{x\to1}\frac{x^{m+1}-x^{n+1}+x^n-mx+m-1}{(x-1)^2}$$ I used substitution $t=x-1 \Rightarrow t\rightarrow 0$ After this, the limit is: ...
-2
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0answers
28 views

A problem of Taylor series [on hold]

I need a step-by-step solution to the following problem. Sorry, I have nothing done because I don't know how to approach the problem. Determine $\alpha \in \mathbb{R}$ such that $$ \arctan^2 (2x) - ...
0
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2answers
32 views

On a certain limit.

Let $\omega$ be an arbitrary irrational number and $ \alpha>0$. Show that the following equality \begin{align*} \lim_{n \to \infty}\frac{\sum_{(i_1,i_2,\cdots, i_n) \in \{1,\cdots,n\}^n} ...
2
votes
0answers
39 views

Lie groups and Lie algebras for matrices

Recently, I stumbled over a few things in very basic Lie group / Lie algebra theory concerning matrix groups. Basically, my question is: Is there a way to canonically understand all the Lie groups ...
0
votes
1answer
24 views

How to check that the function is not absolute continuous

I know when derivative of a function is bounded then the function is absolutely continuous but the converse is not true. How do you check to see a function is not absolutely continuous?.
1
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1answer
13 views

If a homeomorphism between two metric spaces and its inverse are uniformly continuous, do we have “X is bounded implies Y is bounded”?

Let $(X,d_X)$, $(Y,d_Y)$ be metric spaces, $\phi:X\to Y$ be a homeomorphism. Assume $\phi$ and $\phi^{-1}$ are uniformly continuous. Prove or disprove: If $X$ is bounded, then $Y$ ...
0
votes
0answers
14 views

How can we define regular curves implicitly?

Let $F:D\subseteq\mathbb{R}^2\to\mathbb{R}$, $D$ open and connected set, be a $C^1 (D)$ application. What are the minimum requirements for $F$ such that the solutions of the equation $F(x,y)=0$ are ...
1
vote
1answer
27 views

A Banach space in between $L^{1}$ and $L^{2}$, does it make sense?

Let $L^{p} (A, B)$ be a collection of functions $f:A \mapsto B$ satisfying $$(\|f\|_{p})^{p} := \int_{A} |f(x)|^{p} dx <\infty.$$ Now we consider functions $f:[0,1]^{2} \mapsto [0,1]$. We say ...
2
votes
1answer
37 views

Can a function be differentiable at the end points of its (closed interval) domain?

Assume $f$ has a domain of $[a,b]$. Is it possible that $f$ is differentiable on the closed interval $[a,b]$, or must the maximal domain for $f'$ be $(a,b)$?
1
vote
1answer
23 views

Can a function be continuous at the end points of its (closed interval) domain?

Assume $f$ has a domain of $[a, b]$. Is it possible that $f$ is continuous at $x = a$ and $x = b$? If the definition of continuity is that the left and right limits are equal to the function at the ...