The continued fraction of $\pi$ has not yet be known.  I could not see the distribution of the $\{\frac{n!}{2\pi}\}$ where $\{x\}=x-\lfloor x \rfloor$.  Is there any idea how to find two convergent subsequences with different limits?
I encountered a problem today to prove that $(X_n)$ with $X_n = \cos(n!)$ does not have a limit (when $n$ approaches infinity). I have no idea how to do it formally. Could someone help? The simpler the proof (by that I mean less complex theorems are used) the better. Thanks
I would like to use client-side Javascript to perform a DNS lookup (hostname to IP address) as seen from the client's computer. Is that possible? To date I have had an Apple ID tied to my personal email address. It uses my personal paypal account for the purchases, and currently it has a credit balance thanks to some Christmas iTunes gift cards.  With the App Store recently being released, I'd like to buy an OS X app for business purposes. I don't want to use the credit balance on my personal Apple ID account, so I was thinking about creating a new Apple ID just for when I buy apps that I use for business. This way I could have the business Apple ID tied to my business paypal account.  However, I'm concerned that there might be some gotchas with using two different Apple IDs on the same MacBook by the same user. Before I just try this blindly, does anyone have any experience or know of any gotchas? Google page title not matching html  tag Do the electric and magnetic components of an electromagnetic wave really generate each other? Frequently when EM waves are taught, it is said that the change in electric field causes a change in the magnetic field, which then causes a change in the electric field, and so on and so forth.   But from my understanding of basic electromagnetics, it is not necessarily a changing electric field that creates a change in magnetic field, but instead an accelerating charge. and it is this charge that creates both the electric and magnetic field. Again, from my understanding, a changing magnetic field is not generated by a changing electric field, but instead just happens to always be present perpendicular to a changing electric field due to the laws of electromagnetism.   Am I wrong? Or is the "mutual generation" concept between the electric and magnetic components of an EM wave an actual thing? Expressing using properties of vectors It is given in question that $x+a=(a.x)b$ and it is stated that a,b and x are vectors.It is also stated that $a.b$ not equal to $1$.The question ask to express $x$ in term $a$ and $b$,how do you this? The answer is, according to the book: $(m-1)!\cdot \binom{m+r-1}{r}$ I get why this is true. You arrange $m$ people in their seats in $(m-1)!$ ways and then you put $r$ empty spots inbetween them.I was thinking - choose $m$ seats out of $m+r$ to put the people on => $\binom{m+r}{m}$ Arrange them once you've chosen the seats $\to (m-1)!$ So in total: $\binom{m+r}{m} \cdot (m-1)!$ This is obviously not true. What's wrong with the way I'm thinking? How to prefix post permalinks with /blog/ - without affecting custom post types? I have several custom post types, which are properly structured as /%post_type%/%postname%/, however I wish to have the same effect for regular blog entries: /blog/%postname%/  If I prepend the permalink structure with /blog/ it affects the custom post types as well, so currently my permalink structure is simply /%postname%/  Which is the recommended way of doing this? I had a solution involving some rewrites, but the permalink shown in the post editor area does not reflect it. Entering Unicode characters in LaTeX Let $x$ and $y$ be nonzero elements of an integral domain $D$. Then $x$ and $y$ are associates if and only if $x = yd$ for some unit $d \in D$.    I am done proving the $\Leftarrow$ part. For $\Rightarrow$ what I did was: If $x$ and $y$ are associates then $x \mid y$ and $y \mid x$, so $x \mid y$ implies that $xs = y$ for some $s \in D$ and $y \mid x$ implies that $yt = x$ for some $t \in D$. That is, $yts=y$ which implies $ts=1$. Therefore $s$ is a unit and $t$ is a unit.    Am I on the right track? Can I say that the proof is complete? How do I get lightning:datatable to do some undocumented thing X? Canonical Question  I want to do &lt;some really cool thing&gt; with a lightning:datatable, but I can't seem to figure out how to do it. I checked the , but I did not see a way to implement &lt;this really cool feature I need&gt;.  Perhaps I want to provide tooltips, or a pick list, or alter the behavior of the menus, or display images, or show a custom component I created, or define a new row-level click behavior, or alter the keyboard event handlers, etc.  Can I somehow do this? How can i get my community Base URL in apex I am using  Salesforce Community and I have network Id of the community but I don't know how can I get Base URL of my community. Any one know how can I get my community base URL using its id in apex as a internal user.  Is there any other way in which salesforce internal user can get the base URL of all his community in the apex also i don't find any object in which Community Base URL is stored. I checked network sObject but it only store community path prefix not the complete URL so how can i get the base URL for my community? Insert list of figures in the table of contents Is it possible to create a list of all the figures at the end of a document (this I can do) and make it appear in the table of contents?  So in the the table of content there will be a line saying List of figures ... 19 for example? Suppose I sold an option on stock X, with strike price Y, and expiry date Z. Suppose I later buy an option on the same stock, with the same strike price, and the same expiry date. Do these two positions cancel out each other (i.e. both contracts cease to exist anywhere), or will I end up with two separate contracts?  Is it the same with the reverse situation? e.g. I bought an option on stock A, with strike price B, and expiry date C. If I later sell an option on the same stock, with the same strike price and expiry date, will I end up with two contracts (one long position, one short position)? Scale Space Between Multiple Objects If I have multiple objects selected can I use scaling to make the objects closer together (without rescaling the objects)? How can I save application settings in a Windows Forms application? Unable to SSH to my virtual machine via Windows Power Shell ISE The "Satellite Paradox": Twin paradox in orbiting satellites I want to simply eliminate the effect of acceleration, which is the cornerstone in resolving the traditional twin paradox, through the following thought experiment: Assume twins $A$ &amp; $B$ are located in the same orbital height, far away from the surface of a massive planet, on a long concrete column at rest WRT the planet. A rapid explosion that happens between the twins makes them orbit the earth with a high velocity, say, $0.6 \ c$, in the same orbital path but in different directions. According to relativity, since the orbiting satellites are completely inertial frames, each twin attributes a variable time dilation to the other one, and each twin expects the other one to be younger. Nonetheless, these twins are automatically reunited after traveling half of the orbit's circumference without being affected by additional accelerations. Which twin is younger in this case? Indeed, If each twin sees the other's clock as running slow all along the way, how can the clocks agree when they meet? For this configuration:  The acceleration due to the explosion does not affect the twins' clock rates since it occurs abruptly in the same place on the column where the twins are located back-to-back. The clocks have been synchronized just before the explosion.  The traditional gamma factor, attributed to one twin from the viewpoint of the other, is variable because the resultant velocity of one twin complying with the relativistic velocity addition is a variable measured from the viewpoint of the other one. However, whether or not the corresponding time dilation is varying, each twin anticipates the other one to be younger. There is no breaking of symmetry!  The $g$-field of the planet has nothing to do with orbiting satellites since they are inertial, and thus GR corrections are not inserted. (Remember the GPS settings!) How can I completely remove ecryptfs from my system and delete .Private? Few days back I have installed ecryptfs, created a private directory using it. Now my Ubuntu login became super slow. I have tried removing ecryptfs from Synaptic. I found It, it can not be deleted that way. Somewhere I have read about this command  ecryptfs-setup-private --undo   But this command displays a rather confusing list of commands on terminal. I don't want to mess up my installation, neither want to loose my administrator account.  Now my question is, how can I safely remove /home/user/Private (.Private) and remove ecryptfs encryption ? Android and getting a view with id cast as a string We're in the process of deploying a private openstack instance but don't have the Swift object storage service available yet. Can juju bootstrap launch a utility instance without swift? I see that the openstack configuration control-bucket is a container id.