Limit of sequence $ S_n $=$\int_0^1 \frac{nx^{n-1}}{1+x}dx $
Limit of $s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx$ as $n \to \infty$
How to get the day, year, hours, min Individually from date format "yyyy-MM-dd'T'HH:mm:ss.SSSZ"? change citation style ( [1-4] to [1]-[4], and [5, 8-9] to [5], [8]-[9] ) with natbib? I have spent a whole day trying to solve the following problem. Please help if you know how to do this.  I am using natbib (\usepackage[sort&amp;compress,square,comma]{natbib}), and my current citation style looks like [1-4], [5, 8-9].  What is the easiest way to change the citation style to [1]-[4] and [5],[8]-[9]? Knowing the interface of ubuntu and editing them Where are the files for the desktop or the interface that appears on the screen stored? And how can we edit them. How are they programmed?  What programming language is used to built the interfaces? More Question Filtering options I realize that this may be a big request, but...  I think that instead of tabs on the recent question activity pages, or more likely as a new tab, adding a more advanced filtering option.  For instance, being allowed to exclude tags, filter by accepted answers or not, and, most importantly, being able to combine multiple filters!  E.g.: Filtering posts with tag Javascript without tags Jquery and AngularJS with no accepted answers and a score >= 0.  I feel like this is a biggish change that, if implemented as a new tab, wouldn't be that hard to implement, but would be very useful. List all my drafts In GMAIL you can list all your boss-tell-off and wife-cuss-out emails by clicking Drafts    I'd like to see my started-but-never-finished responses, possibly between x Answers and y Votes in /&lt;&lt;user&gt;&gt;/Stats/? Submitting a URL to Google using PHP/cUrl Can we use curved spacetime to measure the one-way speed of light? This is related to  and  The idea is to shine a beam of light from a clock towards a heavy mass, like a black hole, in some manner such that the curvature of spacetime around the mass is sufficient to &quot;bend&quot; the light directly back to the clock. That is, to send the light along a self-intersecting geodesic using a sufficiently curved spacetime. Note that in this case the source and the detector are the same object, so no clock synchronization is required. Hypothetically, this should measure the one-way speed of light. None of the given refutations from the similar posts on this site are convincing. Here are some of the main refutations, as I understand them, and my counter to them: (1) In being curved by spacetime, the light is actually changing direction, hence this is not the one-way speed. To my understanding of General Relativity, the light travels along a geodesic; that is, a curve such that, ignoring technicalities of Differential Geometry, its tangent direction is constant. There is no reflection or acceleration. So in this sense, how can the light with a constant tangent direction be said to change direction? One might say that the path deviates from a straight line in Euclidean space, but given that spacetime is not Euclidean, has no , and has no global notion of direction, this is the only definition of straight or &quot;one direction&quot; that makes sense. Thus, either the light does not change direction and is &quot;one-way,&quot; or we need to be clearer about what exactly we mean by &quot;one-way&quot; in the context of General Relativity. (2) Such an experiment would be impossible or impractical. I think some posters were saying this because the original questions implied that the clock would have to be at the event horizon of a black hole, which is of course impossible or impractical; however, I don't think such an extreme closeness to the black hole is required. Given that the  of a black hole, where light orbits in a circle, is outside of the event horizon, shining the light at some distance outside of the photon sphere should be sufficient to curve the light back to the clock, regardless of the distance of the clock from the black hole (see my rough illustration). At the extreme, we could place the clock directly on the photon sphere, which should be theoretically possible. Regardless, the question is not about practicality, but whether the one-way speed is theoretically measurable.  Update: Essentially what this argument boils down to, I think, is that the average speed of light along a single self-intersecting geodesic may differ from the average speed of light along a non-geodesic self-intersecting path, which is always c. Put that way, however, it does seem a little ridiculous. It is hard to see how the anisotropy of the speed could be defined in such a way as to be consistent with this pathological case. Presumably, we could approximate a single self-intersecting geodesic with a non-geodesic path arbitrarily closely, so, logically, the average speeds over each path should be approximately the same, i.e., c. Furthermore, how would the &quot;preferred direction&quot; of the speed of light behave around a black hole, for instance? Regardless, I still think this experiment does actually solve the problem for the given notion of &quot;direction&quot;. Though the problem is usually posed in the flat spacetime of Special Relativity, with a global notion of direction, so the apparent unanswerability of this question should be seen as a limitation of Special Relativity only, which is then solved by General Relativity. Prove that two distinct numbers of the form $a^{2^{n}} + 1$ and $a^{2^{m}} + 1$ are relatively prime if $a$ is even and have $\gcd=2$ if $a$ is odd.   My attempt: If $a$ is even, let $a = 2^{s}k$ for some integers $k, s$ Then, $$a^{2^{n}} + 1 = 2^{2^{n}s}\cdot k^{2^n} + 1$$ and $$a^{2^{m}}  + 1 = 2^{2^{m}s}\cdot k^{2^m} + 1$$ To prove that they're relatively prime, we need to show that their gcd = 1. And I was stuck here, how could I prove that gcd of two numbers is $1$?     A hint would be sufficient. Thanks. I'm tutoring high school students. I've always taught them that:     A charged particle moving without acceleration produces an electric as well as a magnetic field.   It produces an electric field because it's a charge particle. But when it is at rest, it doesn't produce a magnetic field. All of a sudden when it starts moving, it starts producing a magnetic field. Why? What happens to it when it starts moving? What makes it produce a magnetic field when it starts moving? How do I prove or disprove $X \times (Y \cup Z)= (X \times Y) \cup(X\times Z)$  for all sets $X$, $Y$, and $Z$? I'm lost on the steps here. Theme Activate Hook HDD SMART interpretation - Possible impending drive failure I've just had an issue pop up related to my drives.  First things first, I was working in my case when I clumsily shifted the foot of the case off of my desk. Case shifted and hit the desk somewhat hard, but only fell an inch or two. It had been properly shut down so this may not have anything to do with the issue.  After booting up, I noticed that when playing a video from my F: Drive (Seagate), it produced a whine, almost a chirping sound coming from the hard drive bay. I think I may have heard this before the drop, but quieter.  Fearing impending failure I downloaded GSmartControl and inspected the drive information. The Seagate drive displayed a high Raw Read Error Rate, as well as a high Seek Error Rate.  I've read some of these values can be high on Seagate drives from the start. This drive is about 6 months old and the producer of the noise.  I also checked my Toshiba drive, which came with a refurbished prebuilt I bought about a year ago. It showed more normal SMART data, but displayed some concerning statistics (2 "Mechanical Start Failures" and 2 "Reported Uncorrectable Errors").  Both of the drives passed the basic SMART test, but the noise was quite alarming. It would stop as soon as a paused the video and start right up again as soon as I resumed playing.  Here is an album with 3 photos of the SMART data, the Seagate did not display statistics.  Could I please have help interpreting the SMART data, and whether this may indicate an impending failure?   Thank you for your time.   Here are images of the SMART data. This is problem 3.18 from Patrick Morandi's Field and Galois Theory.     Give an example of fields $k \subseteq K \subseteq L$ and $l \subseteq L$ for which $l/k$ and $L/K$ are algebraic, $k$ is algebraically closed in $K$, and $lK = L$, but $l$ is not algebraically closed in $L$.   This question has been asked twice, namely  and , though neither had any answers, and only some inconclusive comments. Normally I wouldn't ask the question again, but after struggling with it for weeks at this point, I emailed Morandi, and he gave me some hints.  He told me that he remembers finding an example, but that he isn't sure. He told me that it would need to be over characteristic $p$ and that the extensions (at least some of them I suppose) could not be separable. Moreover, he told me that $K = k(x)(a)$, where $a$ is a root of an irreducible polynomial $f(x,y) \in k(x,y)$ (I'm not 100% what the root of a two variable polynomial is). Finally, he said that $l$ is an algebraic extension of $k$, where $f$ becomes reducible over $l$. Still however, I can't come up with anything. I was hoping that with these extra hints, someone would be able to figure it out.  He did point out that there is a chance that there is an error in the question, but I haven't managed to prove that such an example cannot exist. Using memory with a higher MHz rating? Can MacBook Pro's successfully support and fully exploit RAM of a higher MHz rating than that which they ship with?    For example, take my 15" MacBook Pro () that ships with 4GB of 1333MHz DDR3 memory. I'm considering buying 16GB of the same clock speed even though Apple says that it can "support up to 8GB" — .  What about installing 1600MHz in the 16GB variant — will it work? Or will it be clocked down to 1333MHz? When compared to 16GB 1333MHz, will the higher clock speed improve battery life, performance, timings, or what? Why does Ubuntu force users to create a password upon installation? Find $x,y$ given $\gcd(x,y)$ and ${\rm lcm}(x,y)$ These two exercises I encountered recently seem to develop some type of connection between GCD and LCM I can't quite figure out. Exercise 1:  Find all the numbers $x$ and $y$ such that: $a) \ GCD(x,y)=15, \ LCM(x,y)=150$ $b) \ GCD(x,y)=120 \ LCM(x,y)=1320$ $c) \ GCD(x,y)=100 \ LCM(x,y)=990$  Exercise 2:   Find all the numbers $m,n$ such that $GCD(m,n)=pq , \ LCM(m,n)=p^2qs$  where $p,q,s$ are prime  The first thing that is known to me is that $GCD(x,y) \cdot LCM(x,y)= x \cdot y$ Also $LCM(x,y)$ is at most $x \cdot y$ while $GCD(x,y)$ is at most $\max \{x,y\}$. Last thing is that $GCD(x,y)|LCM(x,y)$. Using all this I tried to solve the first exercise: $a)$ First two obvious pairs are $x=15, y=150$ and $y=15, x=150$. Now neither of the numbers can be bigger than $150$ or smaller than $15$. So we are looking for numbers in the range $15-150$ that satisfy $x \cdot y = 15 \cdot 150$ Another such pair is $(x,y)=(30,75), \ (x,y)=(75,30)$. Similarly for $b)$ we find that the only possible values are permutations of the set {$120,1320$} and in $c)$ since $100$ does not divide $990$ no such numbers exist. Now exercise 2 is what made me think there is actually another connection I'm not quite aware of since now it's about arbitrary prime numbers and the previous method doesn't work anymore. My intuition is that it has something to do with $GCD$ or $LCM$ of the $GCD(x,y), \ LCM(x,y)$ Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $ Embedding ordinals in $\mathbb{Q}$ All countable ordinals are embeddable in $\mathbb{Q}$.  For "small" countable ordinals, it is simple to do this explicitly.  $\omega$ is trivial, $\omega+1$ can be e.g. done as $\{\frac{n}{n+1}:n\in \mathbb{N}\} \cup \{1\}$.   $\omega*2$ can be done as $\{\frac{n}{n+1}:n\in \mathbb{N}\} \cup \{1+\frac{n}{n+1}:n\in \mathbb{N}\}$, and $\omega*n$ as $\bigcup_{i\le n} \{i+\frac{n}{n+1}:n\in \mathbb{N}\}$  That immediately also gives $\omega^2$ as  $\bigcup_{i \in \mathbb{N}} \{i+\frac{n}{n+1}:n\in \mathbb{N}\}$  And going further is still relatively easy: We can biject the above embedding of $\omega^2$ onto a single interval $(0,1)$, e.g. through $f(x)=1-\frac{1}{x+1}$ since this is an order preserving bijection from $\mathbb{Q}^+$ to $(0,1)$, allowing us to get $\omega^2+n$, $\omega^2*n$ etc. And by iterating that process, we can get any ordinal below $\omega^\omega$.   But this sort of embedding fails at $\omega^\omega$ as the iteration of $f$ doesn't seem compatible with taking the infinite union - figuratively speaking, we'd squish things to $0$. So what would an explicit embedding of $\omega^\omega$ look like? And the question, then, is if it is possible to give an explicit embedding of $\epsilon_0$? Is the fact that Peano arithmetic cannot prove well-foundedness of $\epsilon_0$ an indication that it cannot be embedded by such elementary functions and their iterations?  And what about even bigger countable ordinals such as the Veblen ordinals, Feferman-Schütte, Bachmann-Howard? From its definition, even if the above all are possible, I assume that the definitive bound of where one can define an effective procedure for the embedding must be Church–Kleene - is that a correct conclusion?  Lastly, does the situation change when we use $\mathbb{R}$ instead of $\mathbb{Q}$, which would allow us to use, perhaps, more easily understood yet inherently complex functions?  Edited because $\omega^2 \neq \omega^\omega$ Suppose there are $2$ permutations given by $p$ and $q$. I need to check whether $p$ belongs to the group generated by $q$, and if so, it's representation in a power of $q$. In other words, I've been given $p$ and $q$, I need to check whether $\exists i \in \{ 0,...,|q|-1 \} \text{ such that } p = q^i$ I know that if $m = \gcd(i, |q|)$, then $|q^i| = \frac{|q|}{m}$. Now, say that such an $i$ exists. Then, $$ |p| = \frac{|q|}{\gcd(i, |q|)} \Rightarrow \gcd(i, |q|) = \frac{|q|}{|p|} $$ Thus, a necessary condition for such an $i$ to exist is: $|p|$ divides $|q|$ Say that this holds, and $\frac{|q|}{|p|} = r$, where r is an integer. Then we just have to solve $\gcd(i, |q|) = r$ for $ i \in \{ 0,...,|q|-1 \}$. Can this be done? If not manually, then by some efficient algorithm? (Note: Just looping over all values of $i$ is not efficient as $i$ may be very large) Edit: Another way I can think of solving this is by writing both $p$ and $q$ as product of disjoint cycles: $$ p = c_1c_2\ldots c_k \text{ and } q = d_1d_2\ldots d_l $$ And thus, as disjoint cycles commute: $$ p^i = c_1^ic_2^i\ldots c_k^i $$ How to proceed after this? How to write below bullet I have to write something under the symbol \bullet in Latex. Is it possible? I'm using the class beamer. Here it is an example of a frame:  \begin{frame} \begin{itemize} \item&lt;1-&gt; Products Market\mbox{}\\ \footnotesize{where $p_{i,t}=u_{i,t}$, with $u_{i,t}\sim u[0,2]$} \vspace{0.5cm} \item&lt;2-&gt; Credit Market \mbox{} \vspace{0.2cm} \\ \footnotesize{where \hspace{0.1cm} $\bullet_{bank} \xrightarrow{loan} \bullet$} \vspace{0.5cm} \item&lt;3-&gt; Interbank Market\mbox{} \vspace{0.08cm} \\ \footnotesize{where \hspace{0.1cm} $\bullet \xrightarrow{loan} \bullet$} \end{itemize} \end{frame}   In particular I want to write "bank" under the bullet, while now it is only in subscript.  Thanks