In many place one finds accounts of how to evaluate $$ \int_0^\infty \frac{\sin x} x\,dx = \underbrace{\lim_{a\to\infty}\int_0^a}_{\text{Why view it this way?}} \frac{\sin x} x\, dx. $$  And it gets asserted that the reason why the integral must first be evaluated over a bounded interval and then afterward the bound is allowed to go to $\infty$ is that $$ \int_0^\infty \left|\frac{\sin x} x\right| \, dx\  \underbrace{{}\  =\infty\  {}}_{\text{That's why.}} \tag 1 $$ and therefore the ways of thinking about integrals introduced by Henri Lebesgue are not applicable on the unbounded interval.  If it is asked how we know that $(1)$ is true, my first thought it that one ought to compare it somehow with a harmonic series because it ultimately declines to $0$ at the same rate.  It seems as if this question is worth having among our stock of questions and answers, but a couple of days ago I found the answer posted as a mere comment.  Hence this present posting.  I'll notify the person who posted the comment.  Doubtless some variety of ways to prove this result exists, so others should post their own versions.
We know that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\int_0^\infty \frac{\sin x}{x} \, dx=\frac{\pi}{2}.$$ How do I show that $$\int_0^\infty \left\vert\frac{\sin x}{x} \right\vert \, dx$$ converges?
Google Drive storage mismatch I have a paid 100GB Google Drive plan. It is reporting that I am at 99% capacity.  According to the the storage both within the desktop and web application, I have used 90.53GB within Drive.    My Drive is mapped to my OS X filesystem, and when fully synched it reports only 61.28GB used.    So what is going on here? I need to do a cleanup but cannot account for the 30GB difference. In the context of topological spaces, I see the following major countability properties:  A space is:   "separable" iff it has a countable dense subset "second countable" iff if has a countable basis "first countable" iff the neighbourhood system of every point has a countable local basis.   (Definitions taken from Counterexamples in Topology by Steen and Seebach, 2nd ed. 1978 -- there may be differences in wording from other sources.)  The question I have is: "separable" into what, exactly? By which I mean to say: what is the thinking behind calling such a condition "separable"?  It appears not to be related to the concept of "separation axioms", which do immediately and obviously invoke an intuitive notation of separation, neither does it seem to have anything to do with "separated sets".  (Anyone using the spelling "seperable" or "seperated" will be immediately downvoted. :-) ) Which indefinite article should precede hour — a or an?        an hour     a hour      Does the usage of an vs a depend on the pronunciation — a history, a hobby, but an hour, an honor? I'm looking to simultaneously send the same email to multiple recipients.  Let's say I want to email John and Bill the same email, I want them to be in the "To:" field, but I don't want them to see each other in the address list.  I don't want to bcc them both because then they see they are bcc'd.  I want them to each get their own copy of the identical email and to have 2 emails in my sent folder - one for each of them.  Essentially I want the same result as if I separately sent them each the identical email, but I'd like to do it in fewer steps.  Can I accomplish this either natively or with a plug-in, or some pre-existing code?  NOTE: I am looking to do this interactively (in real time), not in batch. Shortcut for toggling between different windows of same app? I'm having some trouble understanding cosets.  If I understand it correctly, cosets form a partition. So, if I have $|G:H| = 2$, then $G = H \cup xH$. Right?  In an exercise, I'm asked the following:  "For $G$ group and $H \le G$ such that $|G:H| = 2$, prove that $x^2 \in H$ for any $x \in G$"  If $x \in H$, then $x^2 \in H$ because $H$ is a group. But, if $x \notin H$, then $x \in gH$ for any $g \in G$, right? Then $x^2$ = $(gh)^2$ but I can't see why this is in $H$.  Can you give me some hint? Thanks for stopping by. How to stack three relations Here is my proof, I would appreciate it if someone could critique it for me:  To prove this statement true, we must proof that the two conditional statements ("If $\mathcal{P}(A)⊆ \mathcal{P}(B)$, then $A⊆B$," and, If $A⊆B$, then $\mathcal{P}(A)⊆ \mathcal{P}(B)$) are true.    Contrapositive of the first statement: If $A \nsubseteq B$, then $\mathcal{P}(A) \nsubseteq \mathcal{P}(B)$  If $A \nsubseteq B$, then there must be some element in $A$, call it $x$, that is not in $B$: $x \in A$, and $x \notin B$. Since $x \in A$, then $\{x\} \in \mathcal{P}(A)$; moreover, since $x \notin B$, then $\{x\} \notin \mathcal{P}(B)$, which proves that, if $A \nsubseteq B$, then $\mathcal{P}(A) \nsubseteq \mathcal{P}(B)$. By proving the contrapositive true, the original proposition must be true.    To prove the second statement true, I would implement nearly the same argument, so that isn't necessary to write. So, does this proof seem correct? Also, was the contrapositive necessary? Or is there another way to prove the initial statement? Infinite Cartesian product of countable sets is uncountable Let $\{E_n\}_{n\in\mathbb{N}}$ be a sequence of countable sets and let $S=E_1\times\cdots\times E_n\times\cdots $. Show $S$ is uncountable. Prove that the same statement holds if each $E_n=\{0,1\}$.  By the definition of Cartesian product of sets,   $$\displaystyle S=\Pi_{n\in\mathbb{N}} \{f\colon\mathbb{N}\rightarrow\cup_{n\in\mathbb{N}}E_n\mid\forall n, f(n)\in E_n\}$$  If $E_n=\{ 0,1\}$, then  $$\displaystyle S_{01}=\Pi_{n\in\mathbb{N}}\{0,1\}=E^{\mathbb{N}}$$, where $E=\{0,1\}$.  By a theorem, $\cup_{n\in\mathbb{N}} E_n$ is countable since the sequence is countable.  I'm not sure how to go on from here to show $S$ is uncountable. Can we say anything about the function $f$ that maps a countable $\mathbb{N}$ to another countable union of sequence of countable sets? I'm using Ubuntu 16.04 and there's no fglrx! I'm using Ubuntu 16.04 and there's no fglrx! What happened? What does template  mean? How to see the correctness of close/reopen votes? How can I see the ratio of my previous close/reopen votes that were correct (i.e. resulted in the question being closed/reopened) or incorrect (i.e. resulted in the question left open/closed)?   I am interested in this statistic, because it would help self-educate about the correct use of close/reopen votes. This or That person Today I heard a speaker use the phrase "this person", referring to the subject-person of the story they were telling, and I heard it as (that person). Between the words “this” or “that” in context with the phrase and story-tense, which should you use. Difference of $R^2$ between OLS with individual dummies to panel fixed effect model? Based on my understanding, panel fixed effect model is equivalent to OLS with individual dummies. However, when I ran the two models in R, the $R^2$ from the two models were quite different: 0.8 for OLS with individual dummies and 0.06 for fixed effect model.  Is it the case that in fixed effect model, the fixed effects (individual dummies) are excluded from the calculation of $R^2$? I'm fitting a multiple linear regression model between 4 categorical variables (with 4 levels each) and a numerical output. My dataset has 43 observations.  Regression gives me the following $p$-values from the $t$-test for every slope coefficient: $.15, .67, .27, .02$. Thus, the coefficient for the 4th predictor is significant at $\alpha = .05$ confidence level.  On the other hand, the regression gives me a $p$-value from an overall $F$-test of the null hypothesis that all my slope coefficients are equal to zero. For my dataset, this $p$-value is $.11$.  My question: how should I interpret these results? Which $p$-value should I use and why? Is the coefficient for the 4th variable significantly different from $0$ at the $\alpha = .05$ confidence level?  I've seen a related question, , but there was an opposite situation: high $t$-test $p$-values and low $F$-test $p$-value. Honestly, I don't quite understand why we would need an $F$-test in addition to a $t$-test to see if linear regression coefficients are significantly different from zero. PDOException “could not find driver” Should questions where a problem arose from a typo be closed? Should questions where a problem arose from a typo be closed?  Here is a recent example:    The problem was a misspelled _gaq (written as _gat).   Should these be closed? Surely they're of no use to anyone once the OP recognizes the mistake. If so, what justification should be given? Not a real question? Too localized?     Just to be clear, the question linked above actually was not broken due to a typo. This was my mistake, and raises some other interesting questions. See the discussion below.  If anyone feels like editing this question with examples of actual typo questions, go for it. What is the cardinality of the set of the empty set? Book where wizards learn magic by bonding with crystals that imbue them with powers I read this book a long time ago, but I'm guessing it was written sometime in either the 80's or early 90's. It was a rather long/large book.  Plot summary: Set in a fantasy land of wizards and magic, some kids (maybe) are recruited to learn magic. In the process they find out that wizards are created by being placed in a crashed spaceship that bonds them with crystals that imbue the powers. One of them is able to bond with many/multiple crystals to become very powerful. he became firm friend or friends?