I want to show that every element in the field of fractions of $K[[X]]$ can be written as $\sum_{k=-n}^\infty a_kX^k$ where $a_k\in K, n\in\mathbb{Z}$.  My first attempt was to rewrite  $\displaystyle \frac{\sum_{i=0}^\infty a_iX^i}{\sum_{j=0}^\infty b_jX^j}=\lim_{N\to \infty}\frac{\sum_{i=0}^N a_iX^i}{\sum_{j=0}^N b_jX^j}$  where I am not completely sure, if this is correct. I know that I can write an element of $R[[X]]$ where $R$ is a ring like this. Then I tried polynomial division and some similar things. Also I know that if $b_0$ would be not zero, I can invert the whole polynomial and arrive at the conclusion, even with $n=0$.   However, I was not able to progress from here.
Let $R$ a ring, and let $$\displaystyle R[[x]]=\left\{\sum_{k=0}^{\infty}a_k x^k\;\middle\vert\; a_k\in R\right\}$$ with addition and multiplication as defined for polynomials.  We have that $R[[x]]$ is a ring containing $R[x]$ as a subring.  How to prove that if $a_0\in R$ is a unit, then  $\displaystyle\sum_{k=0}^{\infty}a_k x^k$ is a unit in $R[[x]]$?
Is the sentence "There is a large number of labourers who want to migrate to Japan for work." correct? Possible Duplicate:             There is a large number of labourers who want to migrate to Japan for work.   I type this sentence in a word processor, and it reports an error that "is" should be "are" (subject-verb agreement).  I doubt this is a valid error. Could anyone confirm this? Elements of $\mathbb{F}_p$ having cube roots in $\mathbb{F}_p$ Let $p$ be a prime number, and let $\mathbb{F}_p$ be the field with $p$ elements. How many elements of $\mathbb{F}_p$ have cube roots in $\mathbb{F}_p$?   I had this question on an exam and after reviewing I am still not sure. Any help would be appreciated. On  (but I remember seeing it in other cases) the diff shows the same tag being both added and removed. It is true that $P(X&gt;t+s|X&gt;t)=P(X&gt;s)$ for certain values $t$ and $s$.  However, how can I show that this still holds if:   $T$ is a continuous random variable. That is $P(X&gt;T+s|X&gt;T)=P(X&gt;s)$ Both $T$ and $S$ are continuous RVs: $P(X&gt;T+S|X&gt;T)=P(X&gt;S)$   All the random variables are independent. 32-bit, 64-bit CPU op-mode on Linux I'm confused. Running Fedora Linux, lscpu yields:  Architecture:            i686 CPU op-mode(s):          32-bit, 64-bit ...   But when I try to install a 64-bit program (Chrome) I get error like:     Package /....x86_64.rpm has incompatible architecture x86_64. Valid architectures are ['i686', 'i586', 'i486', i386']   I'm less interested in being able to install Chrome and more interested in why lscpu says that my CPU can run in 64-bit mode; clearly this can't mean I can run 64-bit programs. Can anyone clarify? Why did Superman & Supergirl start to suffocate instantly when Luthor made our yellow Sun red? Spoiler Alert: If you haven't read all comics series &amp; story arc of New Krypton Saga viz. New Krypton, World of New Krypton, Last Stand of New Krypton, Superman: War of Supermen etc, drop this question right away!  While battling the evil alien Brainiac, Superman &amp; Supergirl discovered the long-lost bottled city of Kandor which was stolen from Krypton prior to its destruction. Superman re-enlarged the city on Earth, but it wasn't welcomed by Humans. Humans assaulted the city in which Kryptonian's leader Zor-El (Supergirl's father) was killed. Using highly evolved science, Alura (Zor-El's wife) created a new planet "New Krypton". Plus, she, as a new leader, freed General Zod from Phantom Zone to command new super military of New Krypton (every kryptonians had super powers due to yellow sun).  While New Krypton &amp; Earth were at cold war, Zod ordered his military to attack Earth. When military was in space, Sam Lane (Lois Lane's father) destroyed New Krypton by a trick making their race extinct again. Kryptonian military successfully crushed Earth's Mars defense facility &amp; they headed towards Earth.  When they were about to enter atmosphere of Earth:      Many Kryptonians died under the red sun because they couldn't survive in empty space. Its somewhat plausible because they weren't under yellow sun for long. But, what about Superman &amp; Supergirl? They did survive because they were under yellow sun for very long. But, why did they become very weak instantly? If the power draining rate by radiation of red sun was that high, how did Superman &amp; Supergirl survive?  Is this a plot hole? Or, am I missing something? Can "Mr", "Mrs", etc. be used with a first name? Supersymmetry in Quantum Field Theory I have an idea of supersymmetry in quantum mechanics, can you suggest a book on "supersymmetry in quantum field theory", which has sufficient mathematical rigour like "Peskin and Schroeder" Return a blank when cell reference is blank I am using the following formula in a Google Sheets to calculate the number of days between two dates (dates are in columns G and AL):  =ARRAYFORMULA(if(row(A:A)=1, "Readmission Days", (G:G)-(AL:AL)))   It has to be an array because I am working with a Google Forms and entries are automatically populated into new rows.  The problem is that some of the cells in col AL are blank, so the formula is reading those dates as something else and returning a very large number. Also, when both cells are blank, the formula returns a 0, messing up my data.  How can I include something to ignore blank cells or just return a blank when one or both cell references is blank? Solve $\sum nx^n$ I am trying to find a closed form solution for $\sum_{n\ge0} nx^n\text{, where }\lvert x \rvert&lt;1$.  This solution makes sense to me:  $\sum_{n\ge0} x^n=(1-x)^{-1} \\ \frac{d}{d x} \sum_{n\ge0} x^n = \frac{d}{d x} (1-x)^{-1} \\ \sum_{n\ge0} nx^{n-1} = (1-x)^{-2} \\ x \sum_{n\ge0} n x^{n-1} = x(1-x)^{-2} \\ \sum_{n\ge0} nx^n=\frac x{(1-x)^2}$  However, a  I am reading used the following method:  $$\sum_{n\ge0}nx^n=\sum_{n\ge0}x\frac d{dx}x^n= x\frac d{dx}\sum\limits_{n\ge0}x^n=x\frac d{dx}\frac1{1-x}=\frac x{(1-x)^2}$$  This seems closely related to the solution I described above, but I am having difficulty understanding it.  Can someone explain the method being used here? How do I create subdomain like  ? Do i have to access htaccess somehow? Is it actually simply possible to create it via pure php code or I need to use some external script-server side language?  To those who answered: Well, then, should i ask my hosting if they provide some sort of DNS access?? Previous question:    That was when my buttons died.  On other days, my text dies.    And some times it's just italics.    Now, I like blaming @skiwi for this, and he's often nearby when it happens.  But today, he's not here, and it happened again.    I'm running Windows 8 and using Chrome 40.0.2214.94 m.  Refreshing and such does not fix the issue.  Inspect element -> console -> error log:  http://chat.stackexchange.com/chats/8595/messages/new Failed to load resource: the server responded with a status of 409 (Conflict) http://i.stack.imgur.com/2ZJdJ.png?s=128&amp;g=1&amp;g&amp;s=16 Failed to load resource: the server responded with a status of 502 (Bad Gateway) http://i.stack.imgur.com/boUZp.jpg?g&amp;s=16 Failed to load resource: the server responded with a status of 502 (Bad Gateway) http://i.stack.imgur.com/U8P3C.jpg?s=128&amp;g=1&amp;g&amp;s=32 Failed to load resource: the server responded with a status of 502 (Bad Gateway) http://i.stack.imgur.com/boUZp.jpg?g&amp;s=16 Failed to load resource: the server responded with a status of 504 (Gateway Time-out) http://chat.stackexchange.com/messages/19891816 Failed to load resource: the server responded with a status of 409 (Conflict)   What's going on here?  I should note, I can view italics by selecting, then alt-tabbing:    So it reeks of a client-side issue. What's hiding the text from me?  EDIT:  30 april 2015 it happened again =/ How long will homemade salad dressing stay good in the fridge? How long can I keep salad dressings that I made?  Since this is probably affected by what it's made up of, let's say we're talking about different salad dressings made from a combination of the following ingredients: mayonnaise, ketchup, mustard, vinegar, sugar, garlic, oil, salt, pepper, lemon juice...  And are there any other ingredients that I should watch out for that could affect salad dressing shelf life? Any rules of thumb regarding storing salad dressings...? Reputation misaligned in new topbar In the sites dropdown, the reputation scores are misaligned:    This seems to happen only on Firefox (56.0), both on macOS (10.13) and Windows (7, 10) (thanks @Tom and @MonicaCellio). I've tested a couple of other browsers which do not have this problem. I've already cleared my cache. How "submitted", "to appear", "accepted" papers are evaluated in a CV? Assume a search committee is reading my CV and in the publication section they notice some of my papers are only submitted or claimed to appear in a journal (or accepted for publication in a journal). Sometimes the journal which has accepted the paper for publication lists the title of accepted articles before actually publishing them, but in the rest of cases there is no evidence to prove that the referee process of the paper is over and the journal has accepted the paper for publication. Also assume preprints of my papers are available in ArXiv. So, my questions are:   How a search committee interprets and evaluates those papers which are just submitted or are claimed to appear?  Does a search committee consider these types of publications less valuable than the published ones (assuming the same quality)?  Does a search committee refer to my preprints in ArXiv to evaluate my submitted or accepted papers? How will Ubuntu automatically allocate the disk partition sizes when we select Erase disk and install Ubuntu during installation?    Can the settings for automatic space allocation be changed? Using X11 in OSX 10.8.3 : "ld: library not found for -lX11" A solid sphere has charge $q$ and radius $R$. Find the potential at a point a distance $r$ from the center of the sphere where $r&gt;R$, using infinity as the reference point.  My attempt:  From Gauss' theorem we may deduce that $\displaystyle\mathbf{E}=\frac{1}{4\pi\epsilon_0}\frac{q}{r'^2}\hat{\mathbf{r}}$ where $r'$ is the distance of an arbitrary point from the center of the sphere provided $r'&gt;R$.  $V=-\int_\infty^\mathbf{r}\mathbf{E}\cdot d\mathbf{l}$  My question: What is $d\mathbf{l}$? Since we are traversing in the direction opposite to $\hat{\mathbf{r}}$, I think it should be $-dr'\hat{\mathbf{r}}$. But when I use it to find $V$, I get a sign error. Please help! Homology Groups of $S^n$ We can solve the equation $x^x=c$ using the Lambert $W$ function, but how do we solve $x^{x^x}=c$ ,