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1500 questions
81
votes
12 answers
The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$
What is the sum of the 'second half' of the harmonic series?
$$\lim_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?$$
More precisely, what is the limit of the above sequence of partial sums?
Daniel Pietrobon
- 4,900
81
votes
6 answers
Order of finite fields is $p^n$
Let $F$ be a finite field.
How do I prove that the order of $F$ is always of order $p^n$ where $p$ is prime?
Mohan
- 15,494
81
votes
14 answers
Are all mathematicians human calculators?
I asked my dad why he did not major in math he said "because he is not good at math".
I think I like math, and I think I'm ok at it, but I'm not gifted or anything like that, I just like math. I think I'd like to major in math, but I see all these…
Kat
- 1,371
81
votes
10 answers
Finding out the area of a triangle if the coordinates of the three vertices are given
What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane?
One approach is to find the length of each side from the coordinates given and then apply Heron's formula. Is this the…
TSP1993
- 969
81
votes
1 answer
Compute $ \lim\limits_{n \to \infty }\sin \sin \dots\sin n$
I need your help with evaluating this limit:
$$ \lim_{n \to \infty }\underbrace{\sin \sin \dots\sin}_{\text{$n$ compositions}}\,n,$$
i.e. we apply the $\sin$ function $n$ times.
Thank you.
user6163
81
votes
7 answers
Is there a slowest rate of divergence of a series?
$$f(n)=\sum_{i=1}^n\frac{1}{i}$$
diverges slower than
$$g(n)=\sum_{i=1}^n\frac{1}{\sqrt{i}}$$
, by which I mean $\lim_{n\rightarrow \infty}(g(n)-f(n))=\infty$. Similarly, $\ln(n)$ diverges as fast as $f(n)$, as $\lim_{n \rightarrow…
Meow
- 6,623
81
votes
6 answers
Reference request: introduction to commutative algebra
My goal is to pick up some commutative algebra, ultimately in order to be able to understand algebraic geometry texts like Hartshorne's. Three popular texts are Atiyah-Macdonald, Matsumura (Commutative Ring Theory), and Eisenbud. There are also…
81
votes
4 answers
A simple explanation of eigenvectors and eigenvalues with 'big picture' ideas of why on earth they matter
A number of areas I'm studying in my degree (not a maths degree) involve eigenvalues and eigvenvectors, which have never been properly explained to me. I find it very difficult to understand the explanations given in textbooks and lectures. Does…
robintw
- 963
81
votes
13 answers
What is an example of a sequence which "thins out" and is finite?
When I talk about my research with non-mathematicians who are, however, interested in what I do, I always start by asking them basic questions about the primes. Usually, they start getting reeled in if I ask them if there's infinitely many or not,…
tomos
- 1,801
81
votes
4 answers
What is the difference between the Frobenius norm and the 2-norm of a matrix?
Given a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results?
If they are identical, then I suppose the only difference between them…
Ricket
- 1,181
81
votes
6 answers
How do you compute negative numbers to fractional powers?
My teachers have gone over rules for dealing with fractional exponents. I was just wondering how someone would compute say: $$(-5)^{2/3}$$ I have tried a couple ways to simplify this and I am not sure if the number stays negative or turns into a…
Kot
- 3,323
81
votes
7 answers
What is lost when we move from reals to complex numbers?
As I know when you move to "bigger" number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it hold when you move for example from naturals to…
Юрій Ярош
- 2,034
81
votes
4 answers
Two curious "identities" on $x^x$, $e$, and $\pi$
A numerical calculation on Mathematica shows that
$$I_1=\int_0^1 x^x(1-x)^{1-x}\sin\pi x\,\mathrm dx\approx0.355822$$
and
$$I_2=\int_0^1 x^{-x}(1-x)^{x-1}\sin\pi x\,\mathrm dx\approx1.15573$$
A furthur investigation on OEIS (A019632 and A061382)…
zy_
- 3,031
81
votes
4 answers
$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?
I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this?
Like there is a generic formula for the series:
$$ 1 + 2 + 3 + 4 + \cdots + n =…
vikiiii
- 2,739
- 2
- 35
- 46
81
votes
13 answers
What's a good place to learn Lie groups?
Ok so I read the following article the other day: http://www.aimath.org/E8/ and I wanted to learn more about lie groups. Using my exceptional deduction skills I thought "oh it must have something to do with groups" So I picked up a copy of Dummit…
Asinomás
- 107,565