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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6 views

Check differntiablity of f

Consider a function $ f(x)=|\cos x|+|\sin (2-x)|$ At which of the following points f is not differentiable? a)$\{(2n+1)\frac\pi2:n\in \Bbb Z\}$ b)$\{n\pi:n\in \Bbb Z\}$ c)$\{n\pi+2:n\in \Bbb Z\}$ ...
0
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1answer
19 views

Determine whether the limit exists and justify answer

Usually for these types of questions, I use sequences of functions to show that the limit does not exist, but I don't think I can do this here? I feel like the limit should be zero, but I don't know ...
0
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0answers
13 views

Existence of divergent series $\sum_{n=1}^ \infty a_n$ of decreasing terms whose partial sums are bounded and $\lim (na_n)=0$

Does there exist a decreasing sequence $(a_n)$ of real numbers such that $\lim_{n \to \infty} (na_n)=0$ , the partial sums of $\sum_{n=1}^ \infty a_n$ are bounded , but $\sum_{n=1}^ \infty a_n$ is ...
0
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0answers
11 views

Nonnegative harmonic functions

Suppose $U \in \mathbb{R}^n$ is an open domain, and $u\in C^2(U) \cap C(\bar{U})$ such that $\Delta u = 0$ in $U$. I'm working on a couple of problems pertaining to the mean value formula/harmonic ...
4
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1answer
60 views

find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y \in \mathbb{R}^n$

find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying \begin{equation*} f(x+y)+f(x-y)=2f(x)+2f(y)~\forall x,y \in \mathbb{R}^n. \end{equation*} My attempt: I manage to show ...
1
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1answer
15 views

Characteristic Function in the subset E

Let $E \subseteq \mathbb R$. Then the characteristic function $\chi_{E}:\mathbb R \to \mathbb R$ is continuous if and only if a) $E$ is closed. b) $E$ is Open. c) $E$ is both Open and Closed. d) ...
1
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1answer
25 views

What is the graphical representation of Lipschitz continuity?

I know the graphical represntation for Continuity, Uniform continuity,Absolutely continuity, but I want to know the graphical representation of Lipschitz continuity.
4
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1answer
21 views

Help understanding an inequality on Rudin's construction of the Lebesgue measure

I am having trouble understanding an inequality in Theorem 2.20 from "Real and Complex Analysis." Rudin states that if $f\in\operatorname{C}_c(\mathbb{R}^k)$ , $f$ is real, $W$ is an open k-cell ...
2
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1answer
20 views

Counterexample to the double integral computational theorem when the double integral existence assumption is dropped?

To make things simple, consider the simplest case of the double integral computational theorem. Throughout any phrase involving integrability is in the Riemannian sense. Let $[a, b], [c, d] \subset ...
0
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1answer
21 views

Orthogonal set of a set in Hilbert space

This is an exercise in the Folland Real Analysis p.177. I first thought it is an easy one, but it turns out to be a lot trickier..... I have no idea how to deal with the so-called "double ...
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0answers
13 views

Is it possible to modify norm of Sobolev space suitable for ill-posed problems.

I have trying to pose my problem mathematically for quite some time now. I am not sure even if I am close to defining properly. Would anyone please help: I am interested to study ill-posed problem of ...
2
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1answer
74 views

Calculate $\lim_{n \to \infty}(\sin nx)^\frac{1}{n}$

I know that $$(n)^\frac{1}{n} \to 1$$ and $(a)^\frac{1}{n} \to 1$ (with $a \in \mathbb{R+}$). However, I was wondering what can be said about $$\lim_{n \to \infty }(\sin nx)^\frac{1}{n}$$ and, more ...
2
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6answers
95 views

Calculate the sum of three series which may be telescoping

Let $$\sum_{n=1}^\infty \frac{n-2}{n!}$$ $$\sum_{n=1}^\infty \frac{n+1}{n!}$$ $$\sum_{n=1}^\infty \frac{\sqrt{n+1} -\sqrt n}{\sqrt{n+n^2}}$$ I have to calculate their sums. So I guess they are ...
-3
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1answer
35 views

Does $\sum a_n$ converge if $a_1 = 1$ and $a_{n+1}=\frac{2+\cos n}{\sqrt n}a_n$ [on hold]

Let $$a_1 = 1$$ and $$a_{n+1}=\frac{2+\cos n}{\sqrt n}a_n.$$ Consider $$\sum a_n.$$ How do I calculate if the series converges? The definition by recurrence troubles me a lot.
2
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0answers
33 views

$ \sum_{n=1}^{\infty}\frac{1}{(an^2+b)^k} $ equals $q_0+q_1\pi+\dots+q_k\pi^k$ for nonzero rational coefficients

is it possible to find $a,b\in\mathbb{Z}$ such that for every $k\in\mathbb{N}$ the sum $$ \sum_{n=1}^{\infty}\frac{1}{(an^2+b)^k} $$ equals $q_0+q_1\pi+\dots+q_k\pi^k$ for some nonzero ...
3
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3answers
109 views

Determine if a series defined by cases is convergent and calculate the sum

Consider $\sum_{n=1}^\infty a_n$, where $a_n$ is $$3^{-n}$$ if $n$ is even and $$\ln \frac{(n+2)(n+1)}{n(n+3)}$$ if $n$ is odd. I have to say if it is convergent and calculate its sum, but the ...
6
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1answer
53 views

the series: compute $ \sum_{n=1}^{\infty}\frac{1}{(4n^2-1)^4} $

Compute $$ \sum_{n=1}^{\infty}\frac{1}{(4n^2-1)^4} $$ the result is $\frac{\pi^4+30\pi^2-384}{768}$, so I'm sure the sums $\sum\frac{1}{n^2}$ and $\sum\frac{1}{n^4}$ should appear in the solution. ...
2
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1answer
26 views

If $\frac{n_k}{m_k} \rightarrow \xi \in \mathbb R \setminus \mathbb Q$ then $m_k \rightarrow \infty$

Is it true that if $\frac{n_k}{m_k} \rightarrow \xi \in \mathbb R \setminus \mathbb Q$ then $m_k \rightarrow \infty$? $m_k$ and $n_k$ are integers.
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0answers
20 views

Help in finding a paper on nonlinear Schrodinger equations [on hold]

I'm looking for the following paper: Authors: Baillon, Jean-Bernard; Cazenave, Thierry; Figueira, Mario Title : Equation de Schrodinger non linéaire. (C. R. Acad. Sci. Paris Ser. A-B 284 ...
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2answers
56 views

Find $f$ so that $\int_{1}^{\infty}f(x)dx$ exists, but $\displaystyle \lim_{x\to\infty}f(x)$ does not exist? [duplicate]

I need help finding an example of a function such that $\int_{1}^{\infty}f(x)dx$ converges, but $\displaystyle \lim_{x\to\infty}f(x)$ does not exist. I was trying to find examples of functions ...
0
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0answers
22 views

Proof of $f(x+c) \rightarrow m$ as $x \rightarrow a$

If $f(x) \rightarrow m$ as $x \rightarrow a+c$ then $f(x+c) \rightarrow m$ as $x \rightarrow a$. Proof: Because of the assumption, we have $\forall \varepsilon >0$, there exists $\delta >0$ ...
1
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2answers
27 views

Is this an open set in $\mathbb R^2$?

Is $\{(x,y)\mid y = \sin \frac {1}{x}, x>0\}$ an open set? (It is living in $\mathbb R^2$.) I think it should be open because $(0,0)$ seems to be a limit point of this set while it is not an ...
4
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5answers
71 views

Calculate $\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$

How can I calculate the following limit? I was thinking of applying Cesaro's theorem, but I'm getting nowhere. What should I do? $$\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$$
0
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0answers
16 views

Exactly one supporting line for a $C^1$ Jordan curve

Let $\gamma :[a,b]\to\mathbb{R}^2$ be a convex Jordan curve (closed, simple, continuous) that has $C^1$ regularity, with $\gamma '(t)\neq 0,\ \forall t\in [a,b]$. Prove that there is exactly one ...
0
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2answers
22 views

Continuous and bounded - Check my proof please

Let $f : [0, ∞) → \mathbb{R}$ be continuous such that $\lim_{x→+∞} f(x) = 0$. Prove that $f$ is bounded on $[0, ∞)$ By our hypothesis and the definition of continuity, given $ c \in [0, \infty), ...
2
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0answers
29 views

For which values of $x$ is the following series convergent: $\sum_0^\infty \frac{1}{n^x}\arctan\Bigl(\bigl(\frac{x-4}{x-1}\bigr)^n\Bigr)$

For which values of $x$ is the following series convergent? $$\sum_{n=1}^{\infty} \frac{1}{n^x}\arctan\Biggl(\biggl(\frac{x-4}{x-1}\biggr)^n\Biggr)$$
2
votes
1answer
24 views

Construction of the Itō integral

We fix some filtered probability space $(\Omega,\mathfrak{F},\{\mathfrak{F}_t\}_{t\in[0,T]},\mathbb{P})$. Let, for short, $L^2$ be the space of all progressively measurable processes in ...
2
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0answers
27 views

An exercise about Lebesgue measure

This is part of an exercise from "Real analysis for graduate students" by Richard Bass: Let $m$ be a Lebesgue measure. Suppose for each $n$, $A_n$ is a Lebesgue measurable subset of $[0,1]$. Let $B$ ...
1
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0answers
29 views

Is it Green formulation?

I have this expression: $$-\Delta_p v_n+|v_n|^{p-2}v_n-\theta_n |v_n|^{p^*-2}v_n\rightarrow 0 ~~\text{in}~~ (W^{1,p}_0(\Omega))'$$ It is sying that this expression imply that ...
6
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1answer
61 views

If $f(0)=f(1)=f(2)=0$, $\forall x, \exists c, f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$

Let $f:[0,2]\to \mathbb R$ be a $C^3$ function such that $f(0)=f(1)=f(2)=0$ Prove that $\forall x\in[0,2], \exists c\in[0,2], f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$ This problem got me stuck. I ...
2
votes
1answer
30 views

Uniform Convergence of a Recursive Function

Let $f_{0}$ in $ \mathbb R $ be an increasing and continuous function. Define $f_{n}$ as $f_{n+1}=\arctan\left(x+f_{n}(x)\right)$ for $x$ in $\mathbb R$, $n \geq 1$. Show that $(f_{n})$ converges ...
1
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1answer
32 views

Uniform convergence of recursive sequence

I have the following exercise: Let $\varphi:[0,+\infty)\to \mathbb{R}$ an increasing continuous function that satisfies that $1/2\leq \varphi(x) <1$ for all $x>0$. Let $f_0:[0,+\infty)\to ...
2
votes
1answer
16 views

Pointwise convergence - $\frac{nx}{1+n \sin(x)}$ , $x \in [0, \frac{\pi}{2}]$

Is anyone able to check if this is correct: for $$f_n(x) = \frac{nx}{1+n \sin(x)} , x \in [0, \frac{\pi}{2}]$$ Does this converge pointwise to $$ \frac{x}{\sin(x)}$$ I am unsure due to the fact ...
1
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0answers
30 views

Prove that $g$ is differentiable and that $g'$ is not differentiable at $0$

For part ($c$) I assume I have to show that the limits from above and below are equal? I am having trouble doing this though... I get limit as $h$ tends to $0$ of ( $(4-h^2)^{0.5}$ - ($4^{0.5}$) ...
1
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1answer
19 views

find extremums of $x^2+y^2-12x+16y$ on compact set

I'm trying to find the max/min points of the function \begin{equation*}f(x,y)=x^2+y^2-12x+16y\end{equation*} on the set \begin{equation*}D=\{(x,y):x^2+y^2\leq1\space,\space 3x\geq -y\}\end{equation*} ...
0
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0answers
11 views

unique inner product on a tensor product of Hilbert $C^*$ modules and Hilbert spaces.

For a $C^*-$ algebra $A$ and a Hilbert space $H$ and a Hilbert $A-$module E; how can we show that there is a unique $A-$ valued inner product on $H \otimes E$ as $< h_1 \otimes x_1 , h_2 \otimes ...
0
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0answers
26 views

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

Prove that the following function is $C^\infty$ in the point $\xi=0$: $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ Any ideas how to prove this? I am trying to think ...
0
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0answers
17 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
32 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
30 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 ...
0
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0answers
31 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 - ...
0
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2answers
23 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
42 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 ...
1
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2answers
30 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( ...
0
votes
1answer
38 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 $$\int_a^b f(x)\ \ d(x)=F(b)-F(a)$$
1
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4answers
66 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 ...
1
vote
1answer
26 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
votes
3answers
59 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
votes
1answer
21 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
votes
1answer
67 views

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

Theorem: 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 ...