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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21
votes
2answers
473 views

Closed-form of $\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)=-\int_0^1\frac{\ln(1+x)\ln^3 x}{1-x}\,dx$

Does the following series or integral have a closed-form \begin{equation} \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)=-\int_0^1\frac{\ln(1+x)\ln^3 x}{1-x}\,dx \end{equation} where ...
21
votes
2answers
8k views

Does convergence in $L^{p}$ implies convergence almost everywhere?

If I know $\|f_{n}(x) - f(x)\|_{L^{p}(\mathbb{R})} \rightarrow 0$ as $n \rightarrow \infty$, do I know $\lim_{n \rightarrow \infty}f_{n}(x) = f(x)$ for almost every $x$?
21
votes
2answers
655 views

Compute $\lim\limits_{n\to\infty} \prod\limits_2^n \left(1-\frac1{k^3}\right)$

I've just worked out the limit $\lim\limits_{n\to\infty} \prod\limits_{2}^{n} \left(1-\frac{1}{k^2}\right)$ that is simply solved, and the result is $\frac{1}{2}$. After that, I thought of calculating ...
21
votes
1answer
717 views

Integral $\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}\mathrm dx $

Hi I am trying to evaluate the integral $$ \mathcal{I}(\omega)=\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}\mathrm dx $$ analytically. We can also write $$ ...
21
votes
2answers
555 views

If $f:\mathbb{R}\to\mathbb{R}$ is a left continuous function can the set of discontinuous points of $f$ have positive Lebesgue measure?

If $f:\mathbb{R}\to\mathbb{R}$ is a left continuous function can the set of discontinuous points of $f$ have positive Lebesgue measure? I wondered this today, but made little progress. Thank you.
21
votes
1answer
459 views

Scalar product and uniform convergence of polynomials

Given two functions $u$ and $v$, you can compute $(u|v) = \int_0^1 (u(t)'-u(t))(v(t)'-v(t)) \,\mathrm{d}t$. It resembles the scalar product $\int_{-1}^1 u(t)v(t) \,\mathrm{d}t$, which leads to ...
21
votes
1answer
2k views

Infinite tetration, convergence radius

I got this problem from my teacher as a optional challenge. I am open about this being a given problem, however it is not homework. The problem is stated as follows. Assume we have an infinite ...
21
votes
2answers
787 views

Lebesgue density strictly between 0 and 1

I am having trouble with the following problem: Let $A\subseteq \mathbb{R}$ be measurable, with $\mu(A)>0$ and $\mu(\mathbb{R}\backslash A)>0$. Then how do I show that there exists $x\in ...
20
votes
6answers
3k views

Prove that $\sqrt 2 +\sqrt 3$ is irrational. [duplicate]

Please prove that $\sqrt 2 + \sqrt 3$ is irrational. One of the proofs I've seen goes: If $\sqrt 2 +\sqrt 3$ is rational, then consider $(\sqrt 3 +\sqrt 2)(\sqrt 3 -\sqrt 2)=1$, which implies ...
20
votes
7answers
4k views

Why do introductory real analysis courses teach bottom up?

A big part of introductory real analysis courses is getting intuition for the $\epsilon-\delta$ proofs. For example, these types of proofs come up a lot when studying differentiation, continuity, and ...
20
votes
7answers
2k views

Perfect set without rationals

Give an example of a perfect set in $\mathbb R^n$ that does not contain any of the rationals. (Or prove that it does not exist).
20
votes
15answers
3k views

Useful examples of pathological functions

What are some particularly well-known functions that exhibit pathological behavior at or near at least one value and are particularly useful as examples? For instance, if $f'(a) = b$, then $f(a)$ ...
20
votes
6answers
2k views

Proof that $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$

Thomson et al. provide a proof that $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$ in this book. It has to do with using an inequality that relies on the binomial theorem. I tried to do an alternate proof ...
20
votes
4answers
1k views

Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only

I have found a proof using complex analysis techniques (contour integral, residue theorem, etc.) that shows $$\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$$ for $n\in ...
20
votes
2answers
615 views

The series $\sum_{n=1}^{+\infty}\frac{1}{1^2+2^2+\cdots+n^2}.$

How to justify the convergence and calculate the sum of the series: $$\sum_{n=1}^{+\infty}\frac{1}{1^2+2^2+\cdots+n^2}.$$
20
votes
4answers
1k views

Is there a $\sigma$-algebra on $\mathbb{R}$ strictly between the Borel and Lebesgue algebras?

So, after proving that $\mathfrak{B}(\mathbb{R})\subset \mathfrak{L}(\mathbb{R})$, I asked myself, and now asking you, is there a set $\mathfrak{S}(\mathbb{R})$, which satisfies: ...
20
votes
5answers
1k views

Approximation to $ \sqrt{2}$

I'm a first year Undergraduate student from India. Our professor is going to start a Real Analysis course in September and I was preparing for the initials. I tried and solved many problems, but this ...
20
votes
2answers
2k views

Enumerations of the rationals with summable gaps $(q_i-q_{i-1})^2$

Here is a question from my undergraduate days which I never knew the answer to. I just want to know if anyone can offer me a hint. Consider the rationals in $[0,1]$. Does there exist a (bijective) ...
20
votes
3answers
2k views

$f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$

Trying to solve $f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx $ converges. I need to prove that: $$\lim \limits_{x \to \infty} f(x) = 0$$ Would ...
20
votes
4answers
475 views

Prove that $\int_0^1 \frac{{\rm{Li}}_2(x)\ln(1-x)\ln^2(x)}{x} \,dx=-\frac{\zeta(6)}{3}$

I have spent my holiday on Sunday to crack several integral & series problems and I am having trouble to prove the following integral \begin{equation} \int_0^1 ...
20
votes
6answers
1k views

A question on Terence Tao's representation of Peano Axioms

While reading Terence Tao's book on Analysis I had some questions regarding the implication of the Peano Axioms. After writing the following four axioms (which I will write without changing their ...
20
votes
2answers
491 views

Distance from $x^n$ to lesser polynomials

I am interested in the $L_1$ distance of $x^n$ to the $\mathbb R$-span of $\{1,x,\ldots,x^{n-1}\}$ over some interval. We can WLOG consider the interval $[0,1]$ (say) because scaling and shifting only ...
20
votes
4answers
11k views

max and min versus sup and inf

What is the difference between max, min and sup, inf?
20
votes
1answer
3k views

Example of a Borel set that is neither $F_\sigma$ nor $G_\delta$

I'm looking for subset $A$ of $\mathbb R$ such that $A$ is a Borel set but $A$ is neither $F_\sigma$ nor $G_\delta$.
20
votes
3answers
633 views

Compute $\int_0^\pi\frac{\cos nx}{a^2-2ab\cos x+b^2}\, dx$

How to compute the following integral \begin{equation} \int_0^\pi\frac{\cos nx}{a^2-2ab\cos x+b^2}\, dx \end{equation} I have been given two integral questions by my teacher. I cannot answer ...
20
votes
2answers
446 views

When does $(uv)'=u'v'?$ [duplicate]

In any calculus course, one of the first thing we learn is that $(uv)'=u'v+v'u$ rather than the what I've written in the title. This got me wondering: when is this dream product rule true? There are ...
20
votes
4answers
725 views

How does this discontinuity occur in evaluating a nested square root?

This question is based on a comment I made on a question likely to be closed. Let $$y=\sqrt {x+ \sqrt {x+ \sqrt {x+ \sqrt {x+ \sqrt {x+ \dots}}}}}$$ be the classic nested square root which has ...
20
votes
1answer
525 views

Is there a function having a limit at every point while being nowhere continuous?

Is there a function $\,f:\mathbb{R}\rightarrow\mathbb{R},\,$ that has a limit at every point but is continuous nowhere?
20
votes
3answers
1k views

What's the name of this theorem?

It happens very often in physics that we find relations like: $$\int_V f(x) dx = \int_V g(x) dx$$ for an arbitrary volume $V$. From this we usually say "Since the volume is arbitrary, the integrands ...
20
votes
1answer
405 views

How to find $\int_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}\mathrm dx$

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}\mathrm dx. $$ (I have literature on this, if people want). Note, we can ...
20
votes
4answers
352 views

Continuous $f\colon [0,1]\to \mathbb{R}$ all of whose nonempty fibers are countably infinite?

I have been told it is possible to construct a continuous function $f\colon [0,1]\to \mathbb{R}$ such that $f^{-1}(x)$ is either empty or has cardinality $\aleph_0$ for every $x\in \mathbb{R}$. I've ...
20
votes
4answers
525 views

Evaluating $\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}$

Inspired by Ramanujan's problem and solution of $\sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + \ldots}}}$, I decided to attempt evaluating the infinite radical $$ \sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}} ...
20
votes
1answer
368 views

Learning Aid for Basic Theorems of Topological Vector Spaces in Functional Analysis

I am self-teaching myself the basics of functional analysis (e.g. topological vector spaces), and frankly I am starting to get a migraine sorting out/organizing in my head all of the ...
20
votes
1answer
626 views

real analysis function takes on each value twice?

I want to find a function $f:[0,1] \to [0,1]$ such that $f$ takes on each value in $[0,1]$ exactly twice. I think this means there are an infinite number of discontinuities. Can anyone help me figure ...
20
votes
1answer
286 views

A combination integral and series resulting the inverse tangent integral

$\def\Ti{{\rm{Ti}}_2}$I have been able to solve an integral problem, now I tried to use the other method to crack the integral and I have to prove the following expression \begin{equation} ...
19
votes
4answers
2k views

What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$?

For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. ...
19
votes
4answers
3k views

Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$

While trying some problems along with my friends we had difficulty in this question. True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...
19
votes
4answers
11k views

Discontinuous derivative.

Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is ...
19
votes
5answers
497 views

Computing $\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$

What ways would you propose for the limit below? $$\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$$ Thanks in advance for your suggestions, hints! Sis.
19
votes
2answers
1k views

Is a differentiable function on $(-2, 4)$ always integrable on $[-2, 4]$?

So my question is, say I have a function that is differentiable on $(-2, 4)$. Is it always integrable on $[-2, 4]$? I know that if $f$ is diff on $(-2, 4)$, then it is continuous on $(-2, 4)$. And I ...
19
votes
8answers
505 views

Integral $I:=\int_0^1 \frac{\log^2 x}{x^2-x+1}\mathrm dx=\frac{10\pi^3}{81 \sqrt 3}$

Hi how can we prove this integral below? $$ I:=\int_0^1 \frac{\log^2 x}{x^2-x+1}\mathrm dx=\frac{10\pi^3}{81 \sqrt 3} $$ I tried to use $$ I=\int_0^1 \frac{\log^2x}{1-x(1-x)}\mathrm dx $$ and now ...
19
votes
5answers
5k views

$\epsilon$-$\delta$ proof that $\lim\limits_{x \to 1} \frac{1}{x} = 1$.

I'm starting Spivak's Calculus and finally decided to learn how to write epsilon-delta proofs. I have been working on chapter 5, number 3(ii). The problem, in essence, asks to prove that ...
19
votes
6answers
1k views

$f\colon\mathbb{R}\to\mathbb{R}$ such that $f(x)+f(f(x))=x^2$ for all $x$?

A friend came up with this problem, and we and a few others tried to solve it. It turned out to be really hard, so one of us asked his professor. I came with him, and it took me, him and the ...
19
votes
2answers
1k views

“Pseudo-Cauchy” sequences: are they also Cauchy?

I tried to prove something but I could not, I don't know if it's true or not, but I did not found a counterexample. Let $(a_n)$ be a sequence in a general metric space such that for any fixed $k ...
19
votes
5answers
8k views

What is the 'implicit function theorem'?

Please give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why ...
19
votes
4answers
3k views

Sine function dense in $[-1,1]$

We know that the sine function takes it values between $[-1,1]$. So is the set $$A = \{ \sin{n} \ : \ n \in \mathbb{N}\}$$ dense in $[-1,1]$. Generally, for showing the set is dense, one proceeds, by ...
19
votes
2answers
6k views

Differentiating an Inner Product

If $(V, \langle \cdot, \cdot \rangle)$ is a finite-dimensional inner product space and $f,g : \mathbb{R} \longrightarrow V$ are differentiable functions, a straightforward calculation with components ...
19
votes
2answers
1k views

Tough contest problem

I found this problem in a collection of contest problems of a Russian competition in 1995 and wasn't able to solve it. Solve for real $x$: $$ \cos (\cos (\cos (\cos(x))))=\sin (\sin (\sin (\sin ...
19
votes
2answers
2k views

Olympiad calculus problem

This problem is from a qualifying round in a Colombian math Olympiad, I thought some time about it but didn't make any progress. It is as follows. Given a continuous function $f : [0,1] \to ...
19
votes
5answers
516 views

$f^2+(1+f')^2\leq 1 \implies f=0$

Find all $f\in C^1(\mathbb R,\mathbb R)$ such that $f^2+(1+f')^2\leq 1$ It's quite likely the answer is $f=0$. Note that $|f|\leq 1$ and $-2\leq f'\leq 0$. Therefore $f$ is decreasing and ...