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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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) ...
23
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 ...
23
votes
5answers
8k views

Density of irrationals

I came across the following problem: Show that if $x$ and $y$ are real numbers with $x <y$, then there exists an irrational number $t$ such that $x < t < y$. We know that $y-x>0$. ...
23
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2answers
1k views

Finding the limit of a sequence with an undesirable $\ln k$

I am trying to compute the limit of this sequence: $$\lim\limits_{n \to \infty} \dfrac{(-1)^nn^2}{n!} \sum\limits_{k=2}^{n}\binom{n}{k}(-1)^kk^{n-1}\ln k$$ I can compute without the $\ln k$ in the ...
23
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2answers
797 views

How to prove that $\sum\limits_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum\limits_{k=1}^\infty \frac{1}{(a+k)^2}$ for $a>-1$?

A problem on my (last week's) real analysis homework boiled down to proving that, for $a>-1$, $$\sum_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum_{k=1}^\infty \frac{1}{(a+k)^2}.$$ ...
23
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2answers
2k views

Smallest dense subset of $\mathbb{R}$

I am not sure if what I am looking for even makes sense (or) exists. Anyway I would be happy if someone can clear my confusion. The set of real numbers $\mathbb{R}$ is obtained as completion of ...
23
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2answers
479 views

How to prove $\sum_{n=0}^{\infty} \frac{1}{1+n^2} = \frac{\pi+1}{2}+\frac{\pi}{e^{2\pi}-1}$

How can we prove the following $$\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$$ I tried using partial fraction and the famous result $$\sum_{n=0}^{\infty} ...
23
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1answer
848 views

Is the image of a Borel subset of $[0,1]$ under a differentiable map still a Borel set?

Let $f:[0,1]\to[0,1]$ be a continuous function such that its derivative $f'$ exists on $(0,1)$. Inspired by a similar question of myself here, I want to ask: If $E\subset[0,1]$ is a Borel set, is ...
23
votes
1answer
622 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 ...
23
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1answer
463 views

How should I calculate $\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+…+n^n}{n^n}$

How should I calculate the below limit $$\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+...+n^n}{n^n}$$ I have no idea where to start from.
23
votes
1answer
608 views

Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?

I've written the question first, then the motivation behind it and lastly some background. Note that the question makes references to definitions and theorems written in the background bit at the end. ...
22
votes
4answers
4k 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 $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...
22
votes
3answers
4k views

Proof that the real numbers are countable: Help with why this is wrong

I was just thinking about this recently, and I thought of a possible bijection between the natural numbers and the real numbers. First, take the numbers between zero and one, exclusive. The ...
22
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 ...
22
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 ...
22
votes
3answers
1k views

No continuous function switches $\mathbb{Q}$ and the irrationals

Is there a way to prove the following result using connectedness? Result: Let $J=\mathbb{R} \setminus \mathbb{Q}$ denote the set of irrational numbers. There is no continuous map $f: \mathbb{R} ...
22
votes
2answers
8k 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 ...
22
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4answers
520 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 ...
22
votes
4answers
1k views

Maybe a rather famous integral

How to evaluate : $$\int_0^{\frac{\pi}{2}}\left(\frac{x}{\sin x}\right)^2\text{d}x$$ Thx guys! I was wondering how would use a series expansion?
22
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4answers
879 views

About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $

Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known nice real analysis tools for calculating them (including here all possible ...
22
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4answers
1k views

thoughts about $f(f(x))=e^x$

I was thinking, inspired by mathlinks, precisely from this post, if there exists a continuous real function $f:\mathbb R\to\mathbb R$ such that $$f(f(x))=e^x.$$ However I have not still been able to ...
22
votes
3answers
1k views

Why are every structures I study based on Real number?

I've been studying basic concepts of inner product vector space, normed vector space and metric space. And all the inner products, norms and metrics are defined to be real-valued functions in my ...
22
votes
3answers
567 views

Compute $\sum_{k=0}^{\infty}\frac{1}{2^{k!}}$

How could the series below be computed ? $$\sum_{k=0}^{\infty}\frac{1}{2^{k!}}$$ It's not a series from a book, but a series I thought of many times, and I didn't manage to figure out what I should do ...
22
votes
4answers
661 views

Evaluating $\sum\limits_{n=1}^{\infty} \frac{1}{n\operatorname{ GPF}(n)}$, where $\operatorname{ GPF}(n)$ is the greatest prime factor

$\operatorname{ GPF}(n)=$Greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. How to evaluate convergence/divergence/value of the sum $$\sum_{n=1}^{\infty} ...
22
votes
2answers
650 views

Can we make $\tan(x)$ arbitrarily close to an integer when $x\in \mathbb{Z}$?

My 7-year-old son was staring at the graph of tan() and its endlessly-repeating serpentine strokes on the number line between multiples of $\pi$ and he asked me the question in the title. More ...
22
votes
4answers
14k views

max and min versus sup and inf

What is the difference between max, min and sup, inf?
22
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1answer
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A smooth function's domain of being non-analytic

I am wondering how much a smooth function may be non-analytic, because in proofs, whilst there non-analytic smooth functions, it would suffice if a smooth function were analytic on only a "small set". ...
22
votes
2answers
493 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 ...
22
votes
3answers
948 views

Who came up with the $\varepsilon$-$\delta$ definitions and the axioms in Real Analysis?

I've seen a lot of definitions of notions like boundary points, accumulation points, continuity, etc, and axioms for the set of the real numbers. But I have a hard time accepting these as "true" ...
22
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3answers
770 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 ...
22
votes
1answer
422 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 ...
22
votes
1answer
389 views

The Fibonacci sum $\sum_{n=0}^\infty \frac{1}{F_{2^n}}$ generalized

The evaluation, $$\sum_{n=0}^\infty \frac{1}{F_{2^n}}=\frac{7-\sqrt{5}}{2}=\left(\frac{1-\sqrt{5}}{2}\right)^3+\left(\frac{1+\sqrt{5}}{2}\right)^2$$ was recently asked in a post by Chris here. I ...
22
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3answers
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When L'Hôpital's Rule Fails

I was discussing L'Hôpital's Rule with a Calculus I student earlier today. I mentioned that if the limit obtained by differentiating the numerator and denominator doesn't exist, then L'Hôpital's Rule ...
22
votes
1answer
519 views

If $d$ is a metric, is $d(x,y)/(1+d(x,0)+d(y,0))$ a metric?

I now that one can show that if $d$ is a metric on a vectorspace $X$ then so is $$\varrho(x,y):=\frac{d(x,y)}{1+d(x,y)}.$$ This easily follows from the fact that the function $s \mapsto \frac{s}{1+s}$ ...
22
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1answer
483 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 ...
22
votes
2answers
928 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 ...
22
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1answer
946 views

Expository articles on Analysis and Probability theory

When I come across a notion from algebra or number theory which I don't know I usually check Keith Conrad's page to see if he has written something about it. Key features of his articles are a very ...
21
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13answers
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Intuitive proofs that $\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n=e^x$

At this link someone asked how to prove rigorously that $$ \lim_{n\to\infty}\left(1+\frac xn\right)^n = e^x. $$ What good intuitive arguments exist for this statement? Later ...
21
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7answers
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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).
21
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5answers
514 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.
21
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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 $f(a)$ ...
21
votes
3answers
5k views

Sum of two closed sets in $\mathbb R$ is closed?

Is there a counterexample for the claim in the question subject, that a sum of two closed sets in $\mathbb R$ is closed? If not, how can we prove it? (By sum of sets $X+Y$ I mean the set of all sums ...
21
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3answers
691 views

How to find PV $\int_0^\infty \frac{\log \cos^2 \alpha x}{\beta^2-x^2} \, \mathrm dx=\alpha \pi$

$$ I:=PV\int_0^\infty \frac{\log\left(\cos^2\left(\alpha x\right)\right)}{\beta^2-x^2} \, \mathrm dx=\alpha \pi,\qquad \alpha>0,\ \beta\in \mathbb{R}.$$ I am trying to solve this integral, I ...
21
votes
3answers
3k views

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

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 ...
21
votes
3answers
2k views

Choice of $q$ in Baby Rudin's Example 1.1

First, my apologies if this has already been asked/answered. I wasn't able to find this question via search. My question comes from Rudin's "Princicples of Mathematical Analysis," or "Baby Rudin," ...
21
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6answers
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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 ...
21
votes
4answers
825 views

What are the zero divisors of $C[0,1]$?

Suppose you have a ring $(C[0,1],+,\cdot,0,1)$ of continuous real valued functions on $[0,1]$, with addition defined as $(f+g)(x)=f(x)+g(x)$ and multiplication defined as $(fg)(x)=f(x)g(x)$. I'm ...
21
votes
2answers
510 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 ...
21
votes
2answers
2k views

Better Proofs Than Rudin's For The Inverse And Implicit Function Theorems

I am finding Rudin's proofs of these theorems very non-intuitive and difficult to recall. I can understand and follow both as I work through them, but if you were to ask me a week later to prove one ...
21
votes
6answers
2k views

Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?

I've been interested in non-standard analysis recently. I was reading up on it and noticed the following interesting comment on the Wikipedia page about hyperreal numbers, right after giving an ...