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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4
votes
2answers
214 views

Converse of mean value theorem

I am wondering if the following converse (or modification) of the mean value theorem holds. Suppose $f(\cdot)$ is continuously differentiable on $[a,b]$. Then for all $c \in (a,b)$ there exists $x$ ...
23
votes
4answers
996 views

Integral $\int_0^\infty \log(1+x^2)\frac{\cosh{\frac{\pi x}{2}}}{\sinh^2{\frac{\pi x}{2}}}\mathrm dx=2-\frac{4}{\pi}$

Hi I am trying to show$$ I:=\int_0^\infty \log(1+x^2)\frac{\cosh{\frac{\pi x}{2}}}{\sinh^2{\frac{\pi x}{2}}}\mathrm dx=2-\frac{4}{\pi}. $$ Thank you. What a desirable thing to want to prove! It is a ...
0
votes
0answers
52 views

Minimization of product function subject to constraints

I want to minimize the following function: $\prod_{i=1}^{n}{x_i}$ Subject to the following constraints: $\sum_{i=1}^{n}{x_i}=1.1+(n-1)(0.1)$ and $0.1 \leq x_i \leq 1.1$ How should I go about it? ...
1
vote
1answer
68 views

Product of Limitsuperior of bounded sequences

$\{a_{n}\}$,$\{b_{n}\}$ are two bounded sequences.How can we prove that $\limsup(a_{n}b_{n})$ = $\lim(a_{n})\limsup(b_{n})$. Is it equal to $\lim(a_{n})\lim(b_{n})$? If the sequences are not ...
0
votes
1answer
69 views

point wise and uniform convergence of function series

i need some help to understand point wise and uniformly convergence and solve the following: Let f be a series of functions defined by $f_n(x) := \dfrac{1}{n}e^{-n²x²}$. Show that $f'_n(x)$ ...
5
votes
1answer
302 views

A Riemann integrable function must have infinitely many points of continuity

I was wondering whether anyone would be so kind as to briefly check my proof? I am supposed to prove the statement without using any theorems which would render the proof trivial. If ...
0
votes
2answers
289 views

Maximum value of line integral

I was looking at the following problem: Among all smooth simple closed curve $C$ oriented counterclockwise find the maximum value of $$\int_{C} (4x^2y+y^3)dx+(x^3+4x-xy^2)dy$$. My question is how ...
1
vote
1answer
65 views

$\limsup s_n = \infty$, $\liminf t_n >0$, prove that $\limsup s_n t_n = \infty$

I need some help with the last step. Here's what I have already done. Proof: $\limsup s_n =0$ implies that $\lim_{N\to\infty}\sup\{s_n:n>N\}=\infty$, by definition. We know that ...
2
votes
1answer
68 views

Prove $ \lim_{n \to \infty} \int_a^b \left (1+\frac x n \right)^ne^{-x} dx = b-a $

So I know if you let $f_n(x) = \left(1+\frac x n \right)^ne^{-x}$, then $f(x) = e^{x}\cdot e^{-x} = 1$ Thus, the integral from $\int_a^b dx$ = $b-a$. I'm confused about how we know $f_n$ is ...
1
vote
2answers
85 views

Estimate of L2 norm of a product of functions

Assume that $f,g, f\cdot g\in L^2(\mathbb{R}^n)$. Is the estimate $\|fg\|_{L^2(\mathbb{R}^n)}\leq \|f\|_{L^2(\mathbb{R}^n)}\|g\|_{L^2(\mathbb{R}^n)} $ correct? Thank you!
1
vote
1answer
47 views

Proof concerning logs and taylor series

Prove that if $n$ is a positive integer and $|x| \leq \dfrac{1}{2}n$ then $(i)\quad n\log\left(1+\dfrac{x}{n}\right)=x+Q_{n}(x)$ where $(ii)\quad |Q_{n}(x)|\leq\dfrac{|x|^{2}}{n}$ and deduce ...
1
vote
1answer
47 views

Question about a power series

For what value of $x$ does the series $$\sum_{}^{}\dfrac{(1+x)^n}{n(n-1)}$$ converge? Show that on a certain range of $x$ it determines a differentiable function whose derivative is $\log(-x)$. ...
1
vote
5answers
255 views

Find, with proof, the following limit

$$\lim_{x \to \infty} \, \cos \left(\dfrac{1}{x}\right)^{x} $$ So with this type of limit, does the value cos(1/x) take priority of the power of x as $x \rightarrow \infty$ ? I checked it on wolfram ...
0
votes
1answer
43 views

The extension of functions

Let $f$ be a smooth function defined on $[a,b]$ and $g$ a smooth function defined on $[c,d]$. If $a<b<c<d,f'(x)>0, g'(x)>0$ and $f(b)<g(c)$, then can we find a function $h: \mathbb R ...
11
votes
5answers
371 views
1
vote
1answer
73 views

Root Test and Ratio Test

$$\sum_{n=1}^{\infty}\left(\dfrac{1}{2^n}\right)e^{(-1)^n\sqrt{n}}$$ How do I do the root test for this series? I know that the root test works and that the ratio test does not but how do I show ...
2
votes
1answer
31 views

uniform convergence real analysis [closed]

Find the limit function $$F(x) = \lim_{n\to \infty} F_n(x)$$ and show that the convergence is uniform on closed subsets of S: $$\begin{align} (a) &&& F_n(x) = x^n \sin(nx); && S = ...
0
votes
1answer
45 views

No Continuous Mapping?

Exercise: For [a,b] a nondegenerate closed, bounded interval, show that there is no continuous mapping from $ L^{1}[a,b]$ onto $ L^{\infty}[a,b] $. My initial thought is to prove by contradiction. ...
0
votes
1answer
73 views

Are these $f_n$ equicontinuous?

Let $f_n$ be a sequence of real-valued functions defined on $\mathbb{R}$ satisfying $f_n \to f$ uniformly in the compact subsets of $\mathbb{R}$ $f_n^{-1}$ is bi-Lipschitz $1 \leq (f_n^{-1})'(x) ...
0
votes
1answer
37 views

What is the relation of Lp bound between tightness?

Suppose that $\{f_n\}$ is bounded in $L^p(\mathbb{R})$ and $f_n$ is a sequence of function. Assume that $p$ is larger than 1 but $p$ is not infinite. Is this sequence tight?
1
vote
1answer
88 views

Elementary topology problem

A function $f: \mathbb{R} \to \mathbb{R}$ is said to be bounded at a point $x_0$ provided that there are positive numbers $\varepsilon$ and $M$ so that $|f (x)| < M$ for all $x \in (x_0 - ...
0
votes
1answer
38 views

Converse of existing question on L^p convergence

My question is about this: Convergence in $L^p$ of $f_n$ implies convergence in $L^1$ of $|f_n|^p$ and $f_n^p$ It was shown that the author's question was indeed true by the use of MVT. Is the ...
8
votes
1answer
261 views

Integral $\int_0^{\infty} \frac{\ln \cos^2 x}{x^2}dx=-\pi$

$$ I:=\int_0^{\infty} \frac{\ln \cos^2 x}{x^2}dx=-\pi. $$ Using $2\cos^2 x=1+\cos 2x$ failed me because I ran into two divergent integrals after using $\ln(ab)=\ln a + \ln b$ since I obtained ...
1
vote
1answer
53 views

Show that for $f,g: [0, + \infty) \to \mathbb{R}$ convex functions of Class $C^2$ their product $fg$ is convex

Problem: Let $f,g: [0, + \infty[ \to \mathbb{R}$ be two convex functions of Class $C^2$. Assume that $$ f(0) \geq 0, g(0) \geq 0 \text{ and } f'(0) \geq 0, g'(0) \geq 0 \tag{!}$$ Show that $fg$ ...
3
votes
2answers
241 views

A real continuous periodic function with two incommensurate periods is constant.

I think I have a proof for the statement, but I can't think of a counter-example when $f: \mathbb{R} \to \mathbb{R}$ is not continous. Here's the problem: Let $f: \mathbb{R} \to \mathbb{R}$ be a ...
0
votes
1answer
61 views

Covering a cube with open balls centered at lattice points

I'm trying to prove that given $\epsilon >0$, the balls $B(\epsilon j;\epsilon)$ cover a cube of the form $T = [-b,b]^n$, where $j=(j_1,...,j_n)$ ranges over all integral lattice points of $R^n$ ...
14
votes
2answers
231 views

Integral$\int_0^\infty \ln x\,\exp(-\frac{1+x^4}{2\alpha x^2}) \frac{x^4+\alpha x^2- 1}{x^4}dx$?

I am trying to prove $$ I:=\int_0^\infty \ln x\,\exp\left(-\frac{1+x^4}{2\alpha x^2}\right) \frac{x^4+\alpha x^2- 1}{x^4}dx=\frac{\sqrt{2\alpha^3 \pi}}{2\sqrt[\alpha]e},\qquad \alpha>0. $$ Note: ...
1
vote
1answer
35 views

Proof for estimation lemma

Revising my lecture notes for real analysis - I got stuck with this proof, more specifically I don't really understand where statement (15) comes from. Any help much appreciated!
0
votes
0answers
30 views

Pointwise limit of functions bounded in $L^1$

If $f_n$ is a sequence of functions which converges pointwise to $f$ almost everywhere, and there is a constant $c > 0$ such that $\|f_n\|_{L^1} \leq c$ for all $n$, then is it true that $f \in ...
1
vote
1answer
92 views

Does the series $\sum_{n\ge1}\frac{ln\left(\frac{n+1}n\right)}{\sqrt n}$ converge?

Could you please give me some hint how to decide about convergence of the series $\sum_{n\ge1}\frac{ln\left(\frac{n+1}n\right)}{\sqrt n}$ ? I tried using comparison test: ...
1
vote
1answer
34 views

Relation between convergence of two series

Let $ ( a_n )_{n \in \mathbb {N}} $ a sequence of real numbers such that $ a_n \ge 0 \ \forall n $. If $\sum_{n=1}^{\infty} \dfrac {\sqrt {a_n}}{n} $ converges, is true that $\sum_{n=1}^{\infty} a_n $ ...
0
votes
1answer
19 views

Showing $f(x) = \sum_{q \le x} p(q)^3$ is well-defined and continuous on irrationals

Consider the function $f(x) = \sum_{q \le x} p(q)^3$ on $\mathbb{R}_{\ge 0}$ where $p(q)$ denotes the popcorn function and $q$ denotes a rational. Question 1: Assuming $x > 0$, how do we know that ...
1
vote
2answers
57 views

Is a bounded continuous function defined on $\Bbb R$ differentiable?

Is a bounded continuous function defined on $\Bbb R$ differentiable? Why so? The query is fueled by the following question: Let $f : \Bbb R \rightarrow \Bbb R$ be a bounded continuous function. ...
1
vote
3answers
71 views

Prove $\frac{x}{1+n^2x^2}$ is uniformly convergent

I need to prove that \begin{equation}f_n(x)=\frac{x}{1+n^2x^2}\end{equation} Converges uniformly to $0$. I've tried a solution: (Scratchwork) Want ...
1
vote
1answer
102 views

From finitely additive to countably additive

Given a finitely additive measure $\mu : \mathcal{B} \rightarrow [0,\infty]$. If we want to show that it is also countably additive, finite additivity implies $$\mu(E) \geq \sum_{n=1}^\infty ...
22
votes
1answer
594 views

Is $\pi$ the best constant in this inequality?

Let $E$ be the set of completely monotonous functions on $[0,+\infty)$, that is $f \in C^\infty([0,+\infty))$ and $\forall\, n\geq 0,\forall\, x\geq 0,\quad(-1)^nf^{(n)}(x)\geq 0.$. For $f\in E$ and ...
1
vote
2answers
96 views

Show that f is continuous if graf(f) is compact

The graph of $f\colon M \to \mathbb{R}$ is a set $\{(x,y)\in M\times R \mid y=f(x)\}$. $M$ is a metric space and $\operatorname{graf}(f)$ is given by $\operatorname{graf}(f)=\{(x,y) \in M \times ...
2
votes
2answers
139 views

If M is connected and f is continuous, prove that the graph of f is connected [closed]

The graph of $f: M \to \mathbb R$ is a set $\{(x,y) \in M \times\mathbb R : y = f(x)\}$ Since $M \times\mathbb R$ is a Cartesian product of two metric spaces it has a natural metric. a) If $M$ is ...
1
vote
2answers
35 views

Sequential Compactness in the space $\mathbb R^n$.

I was reading the proof of the following theorem in "Analysis for Applied Mathematics" by Ward Chenney. Then, I started thinking that if $S$ is any subset of the ball in Lemma $1$ above, then $S$ ...
4
votes
2answers
101 views

Show that if $f:\mathbb{R}\to\mathbb{R}$ is a continuous function such that $f(2x)=f(3x)$ for all $x \in \mathbb{R}$, then f is a constant.

Show that if $f:\mathbb{R}\to\mathbb{R}$ is a continuous function such that $f(2x)=f(3x)$ for all $x \in \mathbb{R}$, then f is a constant. My solution: $$f(2x) = f(3x)$$ $$\implies f(x) = ...
1
vote
0answers
20 views

Derivate of a composition

If $f:I\rightarrow\mathbb{R}$ y $f:J\rightarrow\mathbb{R}$ of class $C^n$ such that $f(I)\subset J$ then $g\circ f: I\rightarrow\mathbb{R}$ is of class $C^n$ How do I prove this without using an ...
14
votes
5answers
772 views

Integral $\int_0^{\pi/4} \frac{\ln \tan x}{\cos 2x} dx=-\frac{\pi^2}{8}.$

$$I:=\int_0^{\pi/4} \frac{\ln \tan x}{\cos 2x} dx=-\frac{\pi^2}{8}.$$ I am trying to see nice solutions to this integral. I tried the following $$ I=\int_0^{\pi/4}\frac{\ln \sin x}{\cos 2x} ...
12
votes
4answers
343 views

Integral $\int_0^{\pi/2} \ln(1+\alpha\sin^2 x)\, dx=\pi \ln \frac{1+\sqrt{1+\alpha}}{2}$

$$ I_1:=\int_0^{\pi/2} \ln(1+\alpha\sin^2 x)\, dx=\pi \ln \frac{1+\sqrt{1+\alpha}}{2}, \qquad \alpha \geq -1. $$ I am trying to prove this integral $I_1$. We can write $$ \int_0^{\pi/2} ...
4
votes
2answers
141 views

Closed form of $ \int_0^{\pi/2}\ln\big[1-\cos^2 x(\sin^2\alpha-\sin^2\beta \sin^2 x)\big]dx$

Hello I am trying to solve an incredible integral given by $$ \int_0^{\pi/2}\ln\big[1-\cos^2 x(\sin^2\alpha-\sin^2\beta \sin^2 x)\big]dx=\pi \ln\bigg[\frac{1}{2}\left(\cos^2\alpha +\sqrt{\cos^4 \alpha ...
0
votes
0answers
36 views

How do I convert a function $f(x)$ to $f(\frac{x}{a})$?

I have a pretty simple problem but I kinda am confused. I have the function $$f(x)_a = \sqrt{\frac{2}{\pi}} \cdot \frac{Aa}{x^2 + a^2}$$ And especially: $$f(x=0)_a = \sqrt{\frac{2}{\pi}}\cdot ...
1
vote
0answers
37 views

Prove that limit $ a^x \text{is } a^c$

Prove that $\lim\limits_{x\to c} a^x = a^c$ strictly by the epsilon-delta definition. I know it is quite easy to prove this using logarithm but that is assuming that we already know the ...
0
votes
1answer
432 views

Prove that if f is a continuous strictly monotone function defined on an interval, then its inverse is also a continuous function.

There is a theorem on continuous function that goes as follow: If f is a continuous strictly monotone function defined on an interval, then its inverse is also a continuous function. I have quite an ...
1
vote
1answer
42 views

Two functions intersect,solve the equation.

Given two functions,show where they intersect $(x^2−5)^2/(x+7)^2=\sqrt{169-x^2}$ I have already tried to square both of them but I get a very complex equation and I can not solve it. I saw a guy who ...
0
votes
2answers
67 views

Give an informal reason why this cannot be the gradient of a functoin

Explain why $F(x,y) = \Big(\frac{-y}{x^2 + y^2}, \frac{x}{x^2+y^2}\Big)$ cannot be the gradient of a function (defined away from the origin). Can it be the gradient if we only require F and $f$ to be ...
3
votes
0answers
42 views

If an integral over the plane vanishes, prove that it vanishes on a square.

Let $f\in L^1(\mathbb{R}^2)$ with respect to the Lebesgue measure $m\times m$ on $\mathbb{R}^2$. Prove that if $$\iint_{\mathbb{R}^2} f(x,y)dxdy=0$$ then there exists a square $S_{a,b}=\{(x,y)\,|\, ...