Find limit of $\sqrt[3]{n+1}-\sqrt[3]{n}$
Finding the limit $\lim_{x\rightarrow \infty} \sqrt[3]{x+1}-\sqrt[3]{x}$
"Awesome" vs. "Awful" lstlisting directly after section start removes margin before next section If I place an lstlisting directly after the start of a (sub)section, the margin before the following (sub)section (title) disappears.  Here is a an example document showing the issue:  \documentclass[twocolumn]{article} \usepackage{listings} \begin{document}  \section{Lorem ipsum} Consectetur adipiscing elit.  \begin{lstlisting} Lorem \end{lstlisting}  Sed quis ante nec mi volutpat congue eu id leo.  \section{Dolor sit amet} Suspendisse at lorem mattis, semper massa id, euismod risus.  \newpage  \section{Lorem ipsum} \begin{lstlisting} Lorem \end{lstlisting}  Sed quis ante nec mi volutpat congue eu id leo.  \section{Dolor sit amet} Suspendisse at lorem mattis, semper massa id, euismod risus.  \end{document}   And you can see the (annotated) result here:    Why is there no margin before the fourth section (title)? Is this a bug in my LaTeX environment (MacTeX-2013), or is this the expected behavior? Setting CSS pseudo-class rules from JavaScript I'd like to split strings like these  'foofo21' 'bar432' 'foobar12345'   into  ['foofo', '21'] ['bar', '432'] ['foobar', '12345']   Does somebody know an easy and simple way to do this in python? Propose a privilege to sit in the 500 reputation mark As , the retag privilege serves a lot more confusion and a lot less of its benefit due to the suggested edit system. It must be done away with. It will be done away with. ♪  In light of this decided course of action, we're going to need a new snack for the road between "Create Tags" at 300 reputation and "Established User" at 1000 reputation. We'd like your help in thinking about some ideas on what would be a nice privilege to sit at this level. Since we're... a bit dry on our end.  The primary guideline in deciding what kind of privilege to put here is acknowledging its position in the overall hierarchy of privileges. 500 reputation is an early-intermediate user, one who has started to really get the hang of the site and its function. It can be something that provides convenience in how they currently can act on the site, or it could be some new functionality that fits as an early ability (especially considering there's no new abilities until 2000 reputation's "Edit Everywhat" ability).  I look forward to hearing what creative ideas y'all can provide. I'm on a MacBook Pro (15-inch, 2018) with Mojave 10.14.1. When I connect MBP to a external displays (either via HDMI or via Thunderbolt 2 adapters), the temperature raises sensibly (i.e., CPU temperature lies constantly over 90°) and fans start running fast (i.e., more than 50%) frequently, even if I am doing very simple tasks (e.g., writing in Word with a couple of Chrome tabs open with stackoverflow).  I'm not sure that the problem started with Mojave or if it isn't actually related to the OS version.  I'm wondering if there is a way to limit this annoying overheating as well as fans activity, while using external monitors. Setting up a download page requiring a user to fill a form before downloading anything In the Drupal site I'm setting up, there's a Download page containing several links to download softwares.   What is the best way to "bind" the softwares to the page, put them anywhere online and place a direct link in the page?   Does anyone know a module that would show a contact form when a user clicks one of the "Download" buttons, preventing him from downloading unless he properly completes the form?  Our goal is to receive the contact information of any person downloading our software. Proof that every finite dimensional normed vector space is complete Can you read my proof and tell me if it's correct? Thanks.  Let $V$ be a vector space over a complete topological field say $\mathbb R$ (or $\mathbb C$) with $\dim(V) = n$, base $e_i$ and norm $\|\cdot\|$. Let $v_k$ be a Cauchy sequence w.r.t. $\|\cdot\|$. Since any two norms on a finite dimensional space are equivalent, $\|\cdot\|$ is equivalent to the $l^1$-norm $\|\cdot\|_1$ which means that for some constant $C$, $\varepsilon &gt; 0$, $k,j$ large enough, $$ \varepsilon &gt; \|v_j - v_k\| \geq C \|v_j - v_k\|_1 e_i= C \sum_{i=1}^n |v_{ji} - v_{ki}| \geq |v_{ji} - v_{ki}|$$ for each $1 \leq i \leq n$. Hence $v_{ki}$ is a Cauchy sequence in $\mathbb R$ (or $\mathbb C$) for each $i$. $\mathbb R$ (or $\mathbb C$) is complete hence $v_i = \lim_{k \to \infty} v_{ki} $ is in $\mathbb R$ (or $\Bbb C$) for each $i$. Let $v = (v_1, \dots , v_n) = \sum_i v_i e_i$. Then $v$ is in $V$ and $\|v_k - v\| \to 0$:  Let $\varepsilon &gt; 0$. Then $$ \|v_k - v\| \leq C \|v_k - v\|_1 = C \sum_{i=1}^n |v_{ki} - v_i| \leq C^{'}n \varepsilon$$  for $k$ large enough. How do I write a URL in Markdown on Stack Exchange? Resolving ambiguous overload on function pointer and std::function for a lambda using + $X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed Let $X$ be a topological space. The diagonal of $X \times X$ is the subset $$D = \{(x,x)\in X\times X\mid x \in X\}.$$   Show that $X$ is Hausdorff if and only if $D$ is closed in $X \times X$.   First, I tried to show that $X \times X \setminus D$ is open using the fact that $X \times X$ is Hausdorff (because $X$ is Hausdorff), but I couldn't find an open set that contains a point outside $D$ and is disjoint to it... Apparently recurrences like this cannot be solved with the Master Theorem:  $T(n) = 2T\left(\frac{n}{2}\right) + \frac{n}{\log(n)}$  Because $n^{\log_b(a)} = n^1$ is not a polynomial multiple of $f(n) = \frac{n}{\log(n)}$ since $\frac{n}{\frac{n}{\log(n)}} = \log(n)$, which is less than $n^\epsilon$ for any positive $\epsilon$.  But when is this really even an issue?   For example why wouldn't the time complexity be $n \log n$ using case 2 of the Master Theorem? Unrolling the recursion:  $$T(n) = n\sum_{k=1}^{\log_2(n)} \frac{1}{\log_2(2^k)}$$  $$T(n) = n\sum_{k=1}^{\log_2(n)} \frac{1}{k}$$  Doesn't this suggest $\Theta(n \log n)$ even though it supposedly cannot be done with Master Theorem? How can I permanently enable cheating? Asking feedback/rating for a mobile app? Fig.1: A typical prompt, asking user for feedback/rating in iOS Background Any serious smartphone users must have seen a message similar to Fig.1 above at least once. Variations of such prompt include: &quot;Please give us 5 star ratings if you love this app!&quot;, &quot;Rate us :)&quot;, &quot;Your rating helps improve this app!&quot;, etc. Basically, Fig.1 asks user to leave a (nice) feedback/rating for the app on its AppStore(or equivalents).  The Shorter Question  When and how often should a mobile app user be asked to leave a feedback/ratings?   The Longer Question #1. Which User-statistics? What are the significant user's behavioral statistics that should be based on to generate Fig.1? For example:  Number of runs since install/reinstall of the application Time elapsed since current application run Time elapsed since first application run Time elapsed since first application install Certain events (e.g. after certain number of stages are cleared for a game app)  #2. When to Ask? If multiple user-statistics are chosen to be calculated to generate Fig.1, specifically how should the user-statistics satisfy the condition of the event? For example:  Form a unified equation (e.g. X^2+Y^2=Z^2) with all significant user-statistics partaken fairly to trigger the event. Create (multiple of) &quot;absolute&quot; event triggers (e.g. every 10th app run; elapsing 2 hours since current application run). Both of the two. Generate Fig.1 whichever comes true first. Neither of the two. &quot;There's a better way!&quot;  #3. How often? What would be the optimal min/max frequency of Fig.1 generation? By optimal, I mean users are not to be bothered too much, yet developers gain maximum output of great feedbacks from users? #4. &quot;Don't bother me again :(&quot; Should users have an option to opt-out from receiving Fig.1 at all? #5. Universal User-statistics for All Mobile Apps? Should apps under different categories be based on different user-statistics? There are significant differences in size and type of user-statistics in apps under different categories. For example, a &quot;Flashlight&quot; app is not typically run for one hour straight, while a music/movie streaming app may be run for over one hour. #6. &quot;This app does not need more feedback.&quot; Should every app of all kinds of user base ask this message? If any, should any mobile app stop generating Fig.1 when target conditions are met (e.g. downloads count, active user count, feedback count)? #7. Feedback/Rating Before Deletion Should users be asked to leave a feedback before deleting as seen in Fig.2 below? Users are more likely to leave very critical review on app deletion, but this kind of feedback will be very helpful for the developers to strengthen their app's weakness for next version release.                                                                                                 Fig.2: Asking user for rating before app deletion Prove that:   $$p \,\,\left|\, {p \choose k} \right., \quad 0&lt; k \lt p$$   if $p$ is prime.   how to prove that with direct proof? IIS7 - The request filtering module is configured to deny a request that exceeds the request content length Maximum area of isosceles triangle with perimeter 6 cm Problem. What is the side lengths of an isosceles triangle with $6cm$ perimeter which has the largest area? I solved this problem as below and I would like to learn  that if there is anything wrong: Let us say that the side lengths are $a$ and the bottom length is $6-2a$. By using pythagorean theorem we can conclude that the height $h$ is equal to $\sqrt{6a-9}$. Then, the area of the triangle can be given as $f(a)=(3-a)(\sqrt{6a-9})$. So $f'(a)=-\sqrt{6a-9}+\cfrac{3}{\sqrt{6a-9}}$. Finally, $f'(a)=\cfrac{12-6a}{\sqrt{6a-9}}$ and $f'(a)=0 \iff a =2$. How soon can I re-enter the USA having stayed for 90 days under the Visa Waiver Program? I am currently visiting the USA from the UK (I am a UK citizen) and am intending on staying here for the full 90 days under the Visa Waiver Program.     How long would I have to stay out of the US at the end of this period before returning again?    Would I have to return to the UK, or could I stay with friends in another country (such as Canada) for a while? Is there any way in QGIS to apply service area buffers e.g if I have a point on a map I wish to show the service coverage of 1 mile, 2 mile, 3 mile, breaks using  the road network? Modus Operandi. Formulae for Maximum and Minimum of two numbers with a + b and $|a - b|$ I came across the following problem in my self-study of real analysis:     For any real numbers $a$ and $b$, show that $$\max \{a,b \} = \frac{1}{2}(a+b+|a-b|)$$ and $$\min\{a,b \} = \frac{1}{2}(a+b-|a-b|)$$   So $a \geq b$ iff $a-b \ge0$ and $b \ge a$ iff $b-a \ge 0$. At first glance, it seems like an average of distances. For the first case, go to the point $a+b$, add $|a-b|$ and divide by $2$. Similarly with the second case.   Would you just break it up in cases and verify the formulas? Or do you actually need to come up with the formulas?