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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1
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2answers
22 views

Show that the series converges ($l^2$)

I know that $\sum_{k=1}^\infty|y_n|^2=S<\infty$. I also have that $\lambda >1$. I need to show that $$ \sum_{k=1}^\infty \left| \frac{y_1}{\lambda^k} + ...
0
votes
2answers
28 views

Proving that a sequence converges?

Let $q$, $x$, $y$ be real numbers and $0$$\leq$$q$$<$$1$. Define a sequence ($a_{n}$) by $a_{1}$:=$y$ and $a_{n+1}$:=$q$($a_{n}$+$x$) for all $n$$\geq$$1$. Does ($a_{n}$) converge? So if $q$ is ...
0
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1answer
17 views

Example of equicontinuous sequence of functions which is not convergent

I need to prove there exits an equicontinuous sequence which is not pointwise convergent. I have been working on it but unfortunately , I am not even near to find such sequence of functions. Does ...
1
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0answers
22 views

Calculating in closed form $\int_0^{\infty} \frac{\text{PolyLog}^{(1,0)}(1,-x)}{1+x^2} \, dx$

Can you confirm the following result? Mathematica and other computational stuff I used seem unable to do anything about this result. Maybe to confirm it numerically? $$\int_0^{\infty} ...
1
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1answer
39 views

Evaluation of Real Integral

Given the following definition:$$I=\int\limits_{0}^{2\pi}e^{-i\theta n}\left(\frac{1}{n}\right)^{\rho e^{i\theta}}d\theta$$ Is there an analytic method for evaluating this integral? Best Regards
0
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0answers
5 views

Lipschitz method writing the unique solution.

So the problem gives $f(t,y) = y \cos t$ with $t$ between or equal to $0$ and $2$. I already know the lipschitz method holds with $L=1$. But I'm not sure how to find the unique solution which turned ...
1
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2answers
23 views

Measure space and measurable function

Let $f :\mathbb R\rightarrow \mathbb R$ is a continuous function then the set $\{x \in \mathbb R : \mu ((f^{-1}(x)) >0 \}$ has a zero measure. I think in the case, if f is a step function this ...
0
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0answers
69 views

Conjecturing the closed form $\frac{\pi ^2}{8}-\frac{\pi ^2}{8 \sqrt{2}}+\frac{\pi \log (2)}{4 \sqrt{2}}$

I conjecture that $$\small \int_0^{\pi/2} \frac{\cos ^2(x) \left(-2 \log \left(4^{-\sin ^2(x)} \sin ^{-4 \sin ^2(x)}(x)\right)-4 \log (\cos (x))+\cos (2 x) (4 \log (\cos (x))+\pi +\log ...
2
votes
1answer
41 views

Identity of $\coth $ using Fourier series

The exercise wants me to prove the identity $$\pi \coth \pi a= \frac{1}{a}+ \sum_{n=1}^{\infty}\frac{2a}{n^2+a^2}$$ using the Fourier series of $\cosh ax, \; x \in [-\pi, \pi], \; a \neq 0$. ...
5
votes
2answers
44 views

Are convex functions enough to determine a measure?

Suppose we are talking about $\mathbb{R}^n$. We know that if $\mu$, $\nu$ are two finite Borel measures such that $$\int_{\mathbb{R}^n}f(x) \, d\mu(x)=\int_{\mathbb{R}^n}f(x) \, d\nu(x),$$ for all ...
1
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1answer
28 views

Function on half plane, continuity

Let $\mu$ be a finite positive Borel measure on $\mathbb{R}$ and let $\mathbb{H}$ denote the half plane $\{(x,y) \in \mathbb{R}^2: y > 0\}$. consider the functions ...
0
votes
3answers
36 views

Find bounded function satisfying f(0)=0, f'(0)=0, and bounded first and second derivatives

I am looking for a bounded funtion $f$ on $\mathbb{R}_+$ satisfying $f(0)=0$, $f'(0)=0$ and with bounded first and second derivatives. My intitial idea has been to consider trigonometric functions or ...
0
votes
1answer
19 views

Numerical differentiation (approximation with three supporting points )

Given the supporting points $x-2h,x-h,x+2h$. Determine the difference quotient Du(x) in the form $$Du(x)=au(x-2h)+bu(x-h)+cu(x+2h)$$ for the numerical approximation of $u'(x)$ of order $2$. What ...
0
votes
2answers
45 views

Use Fourier Transform to Show that $f=0$ a.e.

I was working through an old qualifier on my own when I ran across this following question that I was unable to crack. Here it is verbatim: "Let $f\in L^2(\mathbb{R}, \mathcal{L}, m)$ and suppose ...
-1
votes
2answers
44 views

I need help to show that some function is nonnegative

This is a function of $x\in(0,1]$ $$(a_0+v_0 )\left(a_1+\frac{1}{K}\right)\left(a_0+(1-x) \frac{1}{K}\right)-(a_1+v_1 ) \left(a_0+\frac{1}{K}\right)(a_0+(1-x) v_0 )$$ The conditions are: ...
3
votes
4answers
77 views

$f(x) = \frac{2}{x}$ is not uniformly continuous on $(0,1]$

Show that $f(x) = \frac{2}{x}$ is not uniformly continuous on $(0,1]$ WLOG Suppose, $0< \delta \leq 1.$ Let, $\epsilon = 1$ and $x = \frac{\delta}{2}, y = x + \frac{\delta}{3}, x,y \in (0,1]$ ...
1
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1answer
26 views

Rudin 8.16 $\int_X \phi \circ f d\mu = \int_0^\infty \mu\{f > t \} \phi'(t)dt$ hypotheses

Theorem 8.16 in Rudin's Real and Complex analysis states $$\int_X \phi \circ f d\mu = \int_0^\infty \mu\{f > t \} \phi'(t)dt$$ under the conditions that $\mu$ is $\sigma$-finite, $f,\phi \geq 0$ ...
0
votes
1answer
15 views

How to prove this “local invertibility” theorem for bounded linear operators?

The theorem states that, suppose $X,Y$ are complete normed vector spaces, if $\mathscr A_0\in \mathscr L(X;Y)$ is invertible (i.e., $\exists \mathscr A_0^{-1}\in\mathscr L(Y;X)$ s.t. $(\mathscr ...
1
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2answers
37 views

Sum over values of auxiliary function gets arbitrary big, justification

Let $f : \mathbb N_{>0} \to \mathbb R_{\ge 0}$ be a function satisfying $\sum_{n=1}^{\infty} 2^{-f(n)} = \infty$ (like $f(n) = \log n$). Define $$ F(n) = \left\lfloor \log_2\left( \sum_{i=1}^n ...
0
votes
2answers
70 views

Proving that a function is Riemann Integrable

The usual definition to the Riemann integral is: for every $ε>0$, there exists $\delta$ such that if $P$ is a partition of $[a,b]$, and $\|P\|<\delta$, then $|S(f;P)-s|<\epsilon$. Then $f$ is ...
1
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0answers
11 views

Specific problem on Radon measures from Folland's real analysis on $ C_0(X) $

Hello all I am trying to understand the concept of $ C_0(X) $ within the concept of Radon measures as presented in Folland's real analysis chapter 7, so far so good right until I came across problem ...
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votes
0answers
15 views

Csir net question [on hold]

let V be the space of twice differentiable functions on R satisfying f double dash-2f single dash+f=0. define T from V to R(square) by T(f)=(fdash(0),f(0)). Then T is_______ 1) 1-1 and onto 2) 1-1 but ...
3
votes
1answer
45 views

A question on the Banach fixed point theorem.

Suppose $f:(X,\tilde{d})\rightarrow(X,d)$ be a continuous function satisfying \begin{eqnarray}d(f(x),f(y))\leq \lambda d(x,y),\end{eqnarray} $\lambda > 1$. Let $\tilde{d}(x,y)=\lambda d(x,y)$. I ...
0
votes
0answers
42 views

How to prove the following inequality? (or a counter example)

We know that we have $[\int |f(x)|^{p} \mu(dx)]^{1/p}\leq [\int |f(x)|^{q} \mu(dx)]^{1/q}$ when $p\leq q$, where $\mu$ is a probability measure and $f$ is a smooth function. Do we in general have the ...
0
votes
1answer
27 views

Given a real $x$ and an integer $N \gt 1$, prove that there exist integers $h$ and $k$ with $0 \lt k \le N$ such that $|kx-h|\lt 1/N$. [duplicate]

Given a real $x$ and an integer $N \gt 1$, prove that there exist integers $h$ and $k$ with $0 \lt k \le N$ such that $|kx-h|\lt 1/N$. Hint. Consider the $N+1$ numbers $tx-[tx]$ for $t=0,1,2,\dots, N$ ...
2
votes
1answer
36 views

Boundedness of an operator with kernel

Let $K(x,y)$ be measurable in $\mathbb{R}^2.$ Suppose there is a positive, measurable (w.r.t Lebesgue measure on $\mathbb{R}$) $w(x)$ and $A\geq 0$ such that $$\int_{-\infty}^\infty \vert K(x,y) ...
4
votes
3answers
60 views

Determine the value of $ p $ for which the following infinite series converges and for which it diverges.

Determine the value of $ p $ for which the following infinite series converges and for which it diverges: $$ \sum_{n = 2}^{\infty} \frac{\sqrt{n + 2} - \sqrt{n - 2}}{n^{p}}. $$ I don’t know how to ...
1
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6answers
149 views

Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series?

Is there someone who can show me how do I evaluate this sum :$$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$$ Note : In wolfram alpha show this result and in the same time by ratio test ...
-4
votes
2answers
28 views

Let $f$ be a continuous function on $I := [a,b]$, and let $H:I \to \Bbb R$ be defined by $H(x) := \int_x^b f \ \ ,x\in I.$ [on hold]

Let $f$ be a continuous function on $I := [a,b]$, and let $H:I \to \Bbb R$ be defined by $$H(x) := \int_x^b f, \ \ x\in I.$$ To find $H'(x)$ for $x \in I.$ I am stuck with the problem please help.
1
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2answers
67 views

Why do the integers, rationals and any countable set have zero measure?

There is an exercise in my text that tells me to prove the "obvious and easy to see" fact that $\mathbb{Z}$ and $\mathbb{Q}$ have measure zero. Er...here is what I know so far. If I have an interval, ...
3
votes
3answers
84 views

Determine if this series $ \sum_{n=0}^\infty\frac{n^6+13n^5+n+1}{n^7+13n^4+9n+2}$ converges

Determine if the following series converges: $$ \sum_{n=0}^\infty\frac{n^6+13n^5+n+1}{n^7+13n^4+9n+2}. $$ (http://i.stack.imgur.com/qWiuy.png) I don't know how to start.
0
votes
2answers
43 views

Does this function achieve a maximum or minimum?

Suppose that $u$: $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ is continuous and $\lim_{x\to\infty}$ $u(x)$=$\lim_{x\to-\infty}$ $u(x)$=0. Does $u$ achieve a maximum or a minimum value on $\mathbb{R}$? I ...
1
vote
2answers
45 views

Determine if $ \sum_{n \geq 2} \frac{1}{\sqrt[n]{\ln n}}$ congerges

Determine if the following series converges: $$ \sum_{n \geq 2} \frac{1}{\sqrt[n]{\ln n}}.$$ I'm supposed to use here the limit comparison test, but I don't know how to choose the second series.
7
votes
3answers
249 views

What is wrong with this proof of $0=1$?

I am trying to understand what is wrong with the proof posted here that $0=1$ (source): Given any $x$, we have (by using the substitution $u=x^2/y$) ...
0
votes
0answers
28 views

Implict Function Theorem Application

Let $f(x,y)= x^3+y^2-3xy-7=0$ 1)For which points we can find $x=g(y)$ that solves the above 2)Find $g'(y)$ at $x=3,y=4$ 1) By implicit function theorem, for points $(x,y)$ satisfies that ...
2
votes
1answer
110 views

How can I prove the integral $ \int_{1}^{x} \frac{1}{t} \, dt $ is $\ln x $ with this approach?

I have been trying to find a proof for the integral of $ \int_1^x \dfrac{1}{t} \,dt $ being equal to $ \ln \left|x \right| $ from an approach similar to that of the squeeze theorem. Is it possible to ...
0
votes
1answer
19 views

The Riesz transform kernel satisfies Hörmander's condition

Define the kernel $$K_i(x) = \frac{x_i}{\lvert x\rvert^{d+1}}$$ as a function $\mathbb{R}^d \setminus \{ 0 \} \to \mathbb{C}$ for $d \geq 3$ (say). I want to prove that this kernel satisfies ...
0
votes
2answers
27 views

Prove that orthonormalsystem is an orthonormalbasis

We have an orthonormalsystem in $L^2(0, 2\pi)$: $\{e^{ikx} : k \in \mathbb{Z}\}$. Now I want to show that it's also an orthonormalbasis. I thought the easiest way to do that would be to show that ...
0
votes
3answers
43 views

Proof that |x-a| is continuous at x=a (epsilon delta), and nondifferentiable at x=a.

I need help justifying that $|x-a|$ is continuous and non-differentiable at $x=a$. I would also like to prove that it achieves a minimum at $x=a$, but I do not know if that is already clear enough.
0
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2answers
29 views

Totally ordered $\sigma$-algebras

I know that every $\sigma$-algebra is partially ordered with respect to the inclusion operator $\subset$. However, it seems as though every $\sigma$-algebra should be totally ordered with respect to ...
0
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1answer
15 views

Composition of analytic function with arithmetic function

Consider an arithmetic function $g$ with codomain $\{a,b\}$ and a function $f$ which is analytic on some domain including $\{a,b\}$. We therefore have $$f(g(n))=\sum_{k=0}^\infty c_k (g(n)-a)^k$$ and ...
0
votes
2answers
33 views

I need help proving the base case for a mathematical induction proof

I know how mathematical induction works and the generic algorithm of proving a statement by the Principle of Mathematical Induction, but I'm having trouble proving the base case for a particular ...
0
votes
3answers
78 views

why $\frac{t + t^2}{(1 + t)^2} \sim t + t^2$?

While doing my physics homework where answer is given I found that correct answer could be obtained with simplification $\frac{t + t^2}{(1 + t)^2} \sim t + t^2,$ when $t\ll 1$. My attempt was: ...
-8
votes
0answers
24 views

Topic on Measure Theory and Functional Analysis [on hold]

State open mapping theorem and closed graph theorem. Give atleast two examples to show that the closed graph theorem and open mapping theorem may not hold ifthe normed spaces X and Y are not Banach ...
2
votes
1answer
32 views

Compact $K\subset A$ such that $\lambda(K) = \lambda(A) / 2$

Let $A\subset \mathbb{R}$ be a (Lebesgue) measurable set of finite measure. Using the fact that the function $f:\mathbb{R}\rightarrow \mathbb{R}$, $$f(x)=\lambda(A\cap [-x,x]) $$ is continuous, we ...
0
votes
0answers
27 views

Prob. 10, Sec. 4.5 in Kreyszig's Functional Analysis: How to relate this result to solution of equations?

Let $T \colon X \to Y$ be a bounded linear operator, where $X$ and $Y$ are normed spaces, both real or both complex; let $B$ be a subset of the dual space $X^\prime$ (i.e. the normed space of all the ...
1
vote
1answer
61 views

Asymptotic expansion of integral with hyperbolic functions

Consider the integral given by $$f(r)=\int_{0}^{\tanh(r)} \arccos\left(\frac{\sigma}{\sinh(r)\sqrt{1-\sigma^2}}\right)\cdot \frac{1}{\sqrt{\sigma^2+a^2}}d\sigma,$$ where $a>0$. I am wondering ...
0
votes
1answer
23 views

Trace $\sigma$-algebra and measurable envelope

I'm stuck on a problem from Cohn's book. Let $(X,\mathscr{A})$ a measurable space, and let $C$ be a subset of $X$. Let $\mathscr{A}_C$ be the trace of $\mathscr{A}$ on $C$, that is all the ...
6
votes
0answers
121 views

Solving $f'(x) = f(x+1) - f(x)$

Find all $f \in \mathscr{C^\infty}(\mathbb R)$ that satisfy the equation $$f'(x) = f(x+1) - f(x).$$ The 'obvious' answer is the set of all affine maps, but I'm not entirely sure. Some ...
2
votes
0answers
24 views

Show that there is a metric space that has a limit point, and each open disk in it is closed. Collecting examples

This question belongs to the 39th math competitions of Iran. And here is one solution: Suppose that $X=\{\frac{1}{n}: n\in \mathbb{N}\} \cup \{0\}$ and: $d(x,y) = \left\{ \begin{array}{ll} x+y ...