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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
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68
votes
7 answers

Proof that the largest eigenvalue of a stochastic matrix is $1$

The largest eigenvalue of a stochastic matrix (i.e. a matrix whose entries are positive and whose rows add up to $1$) is $1$. Wikipedia marks this as a special case of the Perron-Frobenius theorem, but I wonder if there is a simpler (more direct)…
koletenbert
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68
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2 answers

Is Serge Lang's Algebra still worth reading?

Is Serge Lang's famous book Algebra nowadays still worth reading, or are there other, more modern books which are better suited for an overview over all areas of algebra? EDIT: My main concern is that the first edition of Algebra is already 48 years…
Dominik
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68
votes
4 answers

Algebra: Best mental images

I'm curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on itself. I know that a categorical approach is…
Rachmaninoff
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68
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4 answers

Intuition behind logarithm inequality: $1 - \frac1x \leq \log x \leq x-1$

One of fundamental inequalities on logarithm is: $$ 1 - \frac1x \leq \log x \leq x-1 \quad\text{for all $x > 0$},$$ which you may prefer write in the form of $$ \frac{x}{1+x} \leq \log{(1+x)} \leq x \quad\text{for all $x > -1$}.$$ The upper bound is…
Federico Magallanez
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68
votes
7 answers

Proof that Pi is constant (the same for all circles), without using limits

Is there a proof that the ratio of a circle's diameter and the circumference is the same for all circles, that doesn't involve some kind of limiting process, e.g. a direct geometrical proof?
Chris Card
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68
votes
12 answers

A way to find this shaded area without calculus?

This is a popular problem spreading around. Solve for the shaded reddish/orange area. (more precisely: the area in hex color #FF5600) $ABCD$ is a square with a side of $10$, $APD$ and $CPD$ are semicircles, and $ADQB$ is a quarter circle. The…
Presh
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68
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3 answers

Where does the word "torsion" in algebra come from?

Torsion is used to refer to elements of finite order under some binary operation. It doesn't seem to bear any relation to the ordinary everyday use of the word or with its use in differential geometry (which relates back to the ordinary use of the…
Dan Piponi
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68
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4 answers

Center-commutator duality

I'm reading this article by Keith Conrad, on subgroup series. I'm having trouble with a statement he does at page 6: Any subgroup of $G$ which contains $[G,G]$ is normal in $G$. He says this as evidence that commutator and center play dual roles,…
Bruno Stonek
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68
votes
5 answers

Projection is an open map

Let $X$ and $Y$ be (any) topological spaces. Show that the projection $\pi_1$ : $X\times Y\to X$ is an open map.
Alisha
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68
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3 answers

Is $n \sin n$ dense on the real line?

Is $\{n \sin n | n \in \mathbb{N}\}$ dense on the real line? If so, is $\{n^p \sin n | n \in \mathbb{N}\}$ dense for all $p>0$? This seems much harder than showing that $\sin n$ is dense on [-1,1], which is easy to show. EDIT: This seems a bit…
Per Alexandersson
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68
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5 answers

Why is $A^TA$ invertible if $A$ has independent columns?

How can I understand that $A^TA$ is invertible if $A$ has independent columns? I found a similar question, phrased the other way around, so I tried to use the theorem $$ rank(A^TA) \le min(rank(A^T),rank(A)) $$ Given $rank(A) = rank(A^T) = n$ and…
Chewers Jingoist
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68
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6 answers

Am I too young to learn more advanced math and get a teacher?

I am still 15 years old, but I am very interested in pure math. I have been teaching myself though books, from the internet and from others for the past year or so. I haven't mastered all the topics that are covered in university, just the ones…
Argon
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68
votes
11 answers

Why is it not true that $\int_0^{\pi} \sin(x)\; dx = 0$?

I know the following is not right, but what is the problem. So we want to calculate $$ \int_0^{\pi} \sin(x) \; dx $$ If one does a substitution $u = \sin(x)$, then one gets $$ \int_{\sin(0) = 0}^{\sin(\pi) = 0} \text{something}\; du = 0. $$ We know…
John Doe
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How and why does Grothendieck's work provide tools to attack problems in number theory?

This is probably a horrible question to experts, but I think it is reasonable from someone who knows nothing. I have always been fascinated with Grothendieck and the way he did mathematics. I've heard Mochizuki's work on the abc conjecture heavily…
Arrow
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