Most active questions
460 questions from the last 7 days
40
votes
7
answers
2k
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How far can an infinite number of unit length planks bridge a right-angled gap?
Problem: You are standing at the origin of an infinite flat earth. One quadrant of the earth is gone, leaving an infinite abyss. You have an infinite supply of unit-length zero-width planks. How far ...
7
votes
5
answers
388
views
Prove $\int_0^af(x)\mathrm dx\ge a\int_0^1f(x)\mathrm dx$ for decreasing function
Let $f$ be a decreasing function on $[0,1]$ and $a\in(0,1)$. Prove that
$$\int_0^af(x)\mathrm dx\ge a\int_0^1f(x)\mathrm dx.$$
This would be quite obvious if $f$ were continuous. But for non-...
- 5,070
11
votes
3
answers
2k
views
Can there be a continuous function with infinite derivative everywhere?
Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. For all nowhere-differentiable examples that I know of, for each $a\in\mathbb{R}$ there exist sequences $x_n\to a$ and $y_n\to a$ such that
$$\frac{f(...
- 9,329
17
votes
1
answer
334
views
What is $\lim_n \sqrt[n]{1+\cos(n)}$?
As I was going through some exercise list with limits, I found $\lim_n \sqrt[n]{1+\cos^2(n)}$. This is easy enough, since $\cos^2$ is bounded between 0 and 1, so a squeeze theorem argument lets us ...
- 13.2k
9
votes
3
answers
663
views
Can you prove equality of two expressions by setting them equal in an equation? [closed]
Suppose I have two expressions, and I wish to prove that they are equal to each other. Must I perform algebraic operations on one of the expressions in an attempt to reach the other one? Or perhaps ...
- 93
9
votes
2
answers
415
views
Does this characterize the product topology?
Let $X$ and $Y$ be topological spaces and let $\tau$ be a topology on the set-theoretic product $X \times Y$ such that:
The first projection $p \colon X \times Y \to X$ is continuous and open with ...
- 761
2
votes
2
answers
201
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Is $\varphi = \frac{3\cdot 7\cdot 11\cdot\dots\cdot 599}{5\cdot 9\cdot 13\cdot\dots\cdot 601}$ greater than, less than, or equal to $\frac{1}{13}$?
Is $\varphi = \dfrac{3\cdot 7\cdot 11\cdot\dots\cdot 599}{5\cdot 9\cdot 13\cdot\dots\cdot 601}$ greater than, less than, or equal to $\dfrac{1}{13}$?
My calculator suggests that
$$\ln(\varphi) < -\...
- 1,815
5
votes
1
answer
720
views
Is this a known theorem?
Reference image ^^^
Edit: a person has answered this and I have rediscovered Euclid's Elements, Book 13, Proposition 15.
Ok so I think I might have found a new theorem or maybe rediscovered an old one....
- 103
8
votes
1
answer
455
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A subset of a rectangle that 'blocks' every curve that goes from right to left must connect upper and lower sides
I'm trying to find a proof for the following assetion: Given a rectangular region $R$ and a subset $A$ of $R$, if every curve that starts at the left side of $R$ and ends at the right side intersects $...
- 333
5
votes
6
answers
264
views
Find the ratio $\frac{AC}{BC}$ given a specific configuration of equilateral triangles around a right triangle(need Euclidean geometry approach)
I encountered a geometry problem involving a right-angled triangle and several constructed equilateral triangles. I am trying to solve the second part of the problem (Case 2 in the image).
Continues ...
7
votes
4
answers
489
views
Dummy variable rule for indefinite integrals?
My friend is tutoring high school mathematics, and one of the techniques taught is to let an integral be $I$ then get $I = abc - I$ so that $I = abc/2.$ For example, $$
I := \int e^x\cos{x} dx = eˣ \...
- 463
7
votes
3
answers
257
views
Minimum size of a sequence summing to $2013$ that guarantees a consecutive subset sum of $31$ (still wanted rigorous proof)
I am trying to solve the following problem on integer sequences and subset sums from a 2023 Shanghai high school entrance exam:
Let $A = (a_1, a_2, \dots, a_n)$ be a sequence of positive integers ...
4
votes
6
answers
207
views
Doubt on a proof that $\sin(t)$, $\cos(t)$, and $t$ are linearly independent in the vector space of functions from $\mathbb{R}$ to $\mathbb{R}$
I have this question on a linear algebra worksheet
Let $V$ be the vector space of functions from $\mathbb{R}$ to $\mathbb{R}$. Show that $f,g,h \in V$ are linearly independent, where $f(t)=\sin(t)$, $...
4
votes
2
answers
425
views
Why is the Gödel numbering taken surjective in Ebbinghaus?
It is well known that Gödel numbers are not surjective but injective into $\Bbb{N}$. But on page 182 in Mathematical Logic by Ebbinghaus (2nd edition), it states that the Gödel numbering is surjective....
- 17.4k
2
votes
6
answers
358
views
Find the limit $\lim\limits_{n\to \infty}\sum\limits_{k=1}^{n}\left(\sqrt{1+\frac{k}{n^2}}-1\right)$
I have this limit
$$\lim_{n\to \infty} \sum_{k=1}^{n} \left(\sqrt{1+\frac{k}{n^2}}-1\right)$$ and and I tried to evaluate it in the following way:
First I set $x=\frac{k}{n^2}$ to make the expression ...
- 359