Mathematics

Most Popular

more tags
1500 questions
68
votes
4 answers

Similar matrices have the same eigenvalues with the same geometric multiplicity

Suppose $A$ and $B$ are similar matrices. Show that $A$ and $B$ have the same eigenvalues with the same geometric multiplicities. Similar matrices : Suppose $A$ and $B$ are $n\times n$ matrices over $\mathbb R$ or $\mathbb C$. We say $A$ and $B$ are…
user2723
68
votes
3 answers

Is $ \sum\limits_{n=1}^\infty \frac{|\sin n|^n}n$ convergent？

Is the series $$ \sum_{n=1}^\infty \frac{|\sin n|^n}n\tag{1}$$ convergent？ If one want to use Abel's test, is $$ \sum_{n=1}^\infty |\sin n|^n\tag{2}$$ convergent？ Thank you very much
2016
  • 1,093
68
votes
9 answers

When to give up on a hard math problem?

I practice olympiad problems from books like Putnam and Beyond. Often I come across a problem that I simply can't solve. After $\sim30$ minutes of deep thinking it feels like I'm ramming my head into a brick wall, since I've exhausted all avenues of…
user1299784
  • 2,089
  • 1
  • 21
  • 29
68
votes
6 answers

Can someone please explain the Riemann Hypothesis to me... in English?

I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?
Letseatlunch
  • 839
68
votes
13 answers

Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $

I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges, but what about the alternating harmonic series…
Isaac
  • 37,127
68
votes
2 answers

Category-theoretic limit related to topological limit?

Is there any connection between category-theoretic term 'limit' (=universal cone) over diagram, and topological term 'limit point' of a sequence, function, net...? To be more precise, is there a category-theoretic setting of some non-trivial…
Rafael Mrden
  • 2,289
68
votes
5 answers

Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even

I have a question: Suppose $f$ is continuous and even on $[-a,a]$, $a>0$ then prove that $$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$ How can I do this? Don't know how to start.
James
  • 699
68
votes
4 answers

Integral $\int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx$

Regarding this problem , I conjectured that $$ I(r, s) = \int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx = 4 \pi \operatorname{arccot} \sqrt{ \frac{2r + 2\sqrt{r^{2} -…
Sangchul Lee
  • 186,861
68
votes
9 answers

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real numbers, arguably one of the most beautiful ideas…
Keshav Srinivasan
  • 11,002
68
votes
1 answer

Formal definition of conditional probability

It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space $ (\Omega, \mathscr{A}, \mu ) $ with $\mu(\Omega) = 1 $, and a random variable $ X :…
smiley06
  • 4,367
68
votes
2 answers

Is this similarity to the Fourier transform of the von Mangoldt function real?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: $$\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n}…
Mats Granvik
  • 7,644
68
votes
1 answer

Evaluating $\sum\limits_{x=2}^\infty \frac{1}{!x}$ in exact form.

Introduction: We know that: $$\sum_{x=0}^\infty \frac{1}{x!}=e$$ But what if we replaced $x!$ with $!x$ also called the subfactorial function also called the $x$ th derangement number? This interestingly is just a multiple of $e$ and an Incomplete…
Тyma Gaidash
  • 14,281
68
votes
28 answers

Is there a great mathematical example for a 12-year-old?

I've just been working with my 12-year-old daughter on Cantor's diagonal argument, and countable and uncountable sets. Why? Because the maths department at her school is outrageously good, and set her the task of researching a mathematician, and…
Mark Bennet
  • 102,257
68
votes
7 answers

Is there any geometric intuition for the factorials in Taylor expansions?

Given a smooth real function $f$ , we can approximate it as a sum of polynomials as $$f(x+h)=f(x)+h f'(x) + \frac{h^2}{2!} f''(x)+ \dotsb = \sum_{k=0}^n \frac{h^k}{k!} f^{(k)}(x) + h^n R_n(h),$$ where $\lim_{h\to0} R_n(h)=0$ . There are multiple…
glS
  • 8,094
68
votes
2 answers

Why doesn't Cantor's diagonal argument also apply to natural numbers?

In my understanding of Cantor's diagonal argument, we start by representing each of a set of real numbers as an infinite bit string. My question is: why can't we begin by representing each natural number as an infinite bit string? So that 0 =…
usul
  • 4,104
1 2 3
…
99 100