Characterize units in formal power series $R[[x]]$ Suppose $R$ is a commutative ring with unity. Define $R[[x]]$ as "formal power series in the variable $x$ with coefficients from $R$". These are the infinite sums of the form  $     \sum_{n=0}^\infty a_ix^i,  a_i\in R. $ Is there any way to characterize all of the units in this ring $R[[x]]$?
Prove $\alpha \in R[[x]]$ is a unit iff $a_0 \in R$ is a unit Prove $\alpha \in R[[x]]$ is a unit iff $a_0 \in R$ is a unit.   $"\Rightarrow"$ suppose $\alpha \in R[[x]]$ is a unit, where $\alpha = \sum_{n=0}^{\infty} a_nx^n$  then there exists an inverse $\beta = \sum_{n=0}^{\infty} b_nx^n$   stuck at this spot,  $(\sum_{n=0}^{\infty} a_nx^n)(\sum_{n=0}^{\infty} b_nx^n) = \sum_{n=0}^{\infty}(\sum_{k=0}^{n}(a_kb_{n-k}))x^n$  A little mixed up on how this becomes $1 = \sum_{n=0}^{\infty}e_nx^n$ where $e_0 =1, e_{n&gt;0} = 0$ and why that would make $a_0b_0 = 1$ which would make $a_0 \in R$ a unit. I haven't tried the other direction yet.  EDIT  Maybe I get it, because we assumed this sum is a unit then its inverse exists and so the $x^0=1$ and all $a_ib_i=1$ thus $a_0b_0=1$ which gives $a_0$ is a unit in $R$
Auto-sort column Google Spreadsheet on incoming Google Form response How can I get the Google Spreadsheet to automatically order the incoming form responses alphabetically (2nd column)?   I know how to do this manually or by going in and editing a cell, but cannot get it to do this as an onevent for incoming response. half sine and half cosine quaternions Why is Main method private? How do I delete multiple rows in Entity Framework (without foreach) I have a Python function that takes a numeric argument that must be an integer in order for it behave correctly. What is the preferred way of verifying this in Python?  My first reaction is to do something like this:  def isInteger(n):     return int(n) == n   But I can't help thinking that this is 1) expensive 2) ugly and 3) subject to the tender mercies of machine epsilon.   Does Python provide any native means of type checking variables? Or is this considered to be a violation of the language's dynamically typed design?  EDIT: since a number of people have asked - the application in question works with IPv4 prefixes, sourcing data from flat text files. If any input is parsed into a float, that record should be viewed as malformed and ignored. A contractive mapping on $M$ is a function $f$ from the metric space $(M,d)$ into itself satisfying $$d(f(x),f(y))&lt;d(x,y)$$ whenever $x,y\in M$ with $x\ne y$. Prove that if $f$ is a contractive mapping on a compact metric space $M$, there exists a unique point $x\in M$ with $f(x)=x$.   Consider the function $g(x)=d(f(x),x)$. Note that   \begin{align*}d(f(x),x)-d(f(y),y)&amp;\leq (d(x,y)+d(y,f(x))-(d(y,f(x)-d(f(x),f(y)) \\ &amp;=d(x,y)+d(f(x),f(y)) \\ &amp;&lt;2d(x,y)  \end{align*}  And similarly for $d(f(y),y)-d(f(x),x)$, so we have $|d(f(x),x)-d(f(y),y)|&lt;2d(x,y)$. This means $g(x)$ is continuous.  So $g(x)$ is a real-valued continuous function on a compact space, so must be bounded and has a minimum $c$, i.e. there exists $x_0$ such that $d(f(x_0),x_0)$ equals the minimum $c$. What then? Difference between "speed up something" and "speed something up" Both seem to be in common usage, but is there a nuanced difference about when one is more appropriate than the other? Thanks. So to help prevent brute force attacks against my phpmyadmin install I've configured nginx to require HTTP Basic Auth (extra username and password) and then the requests need to come from a whitelisted IP Address. However the IP Address whitelist works fine if I run  however without adding the extra .php block I could load  without my IP being whitelisted. This issue doesn't affect the HTTP Basic Auth. Is there a reason this happens? Anyway to get around having to add extra .php blocks? A copy of the config area for phpmyadmin is below.  # Setup and secure phpMyAdmin location /phpmyadmin/ {   allow 1.2.3.4;   deny all;   auth_basic "phpMyAdmin - HTTP Basic Login";       auth_basic_user_file /etc/nginx/pma_pass;     location ~ \.php$ {     include snippets/fastcgi-php.conf;             fastcgi_pass unix:/run/php/php7.0-fpm.sock;     allow 1.2.3.4;     deny all;     } } I have a seven hour layover at Brussels airport, and I'd like to know whether there is enough time to leave the airport and go into the Brussels city center for a beer or a meal?  If so, what transportation is available, how does it work, how long does it take, and how much does it cost?  Are there convenient luggage lockers in the airport so that I can avoid taking my carry-on luggage into the city?  This layover is from 8:30 in the morning to 15:30 in the afternoon on a Monday, and the flight is arriving from the United States.  The outgoing flight is within Western Europe. How can I wrap text around a bottom-right div? Can you find all classes in a package using reflection? Is it possible to find all classes or interfaces in a given package? (Quickly looking at e.g. , it would seem like no.) Consider a stochastic process $\{X_t, t = 1, 2, \ldots\}$ following the model $$X_t = \alpha X_{t-1} + e_t,$$ where $e_t \thicksim f$.  Can I say that the distribution of the initial point, $X_1$, is the same as $f$? Can I say that the stationary marginal density, if it exists, of $\{X_t\}$ is same as $X_2 (\stackrel{D}{=}\alpha X_1 + e_2)$?  I think that the stationary marginal density, if it exists, of $\{X_t\}$ is the same as $X_2$, but not necessarily same as $X_1$. It's been more than a month since I have started using . It has these users with user names in the format of "user-random number". I have observed them in some of the stack activities, most of the time asking questions.   Does the moderators create them for their own purpose. If yes what purpose do they serve?   I mean a user is free to have any user name. But it's just a hunch that such users are actually moderators. I may be wrong; just need some clarification. How to display all the tables with more information (create date, size,...) in a MySQL database? How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$? What is the difference between a directory name that ends with a slash and one that does not? Is there any difference between a directory name such as  mydirectory   and  mydirectory/   I noticed this happens when I execute ls in some directories - some of the directory names have a slash and some don't. This is problematic because if I want to access a file contained in a directory, I may need to add a slash at the end:  vi $mydirectory"/"$myfile   or simply do   vi $mydirectory$myfile Adding word 'Table' before each entry in list of tables I need to prefix each entry in list of tables and figures with appropriate text - table or figure for each. Also I need to remove chapter numbers from list, because tables are numbered consequently through the whole report.  I managed to do everyting I need with this simple commands:  \renewcommand{\thetable}{\arabic{table}} \renewcommand{\thefigure}{\arabic{figure}}  \makeatletter \providecommand\phantomsection{}% for hyperref  \newcommand\listoftablesandfigures{%     \chapter*{List of Tables and Figures}%     \phantomsection %\addcontentsline{toc}{chapter}{List of Illustrations}% %\section*{Figures}% %\phantomsection %\addcontentsline{toc}{section}{\protect\numberline{}Figures}% \@starttoc{lof}% \bigskip %\section*{Tables}% %\phantomsection %\addcontentsline{toc}{section}{\protect\numberline{}Tables}% \@starttoc{lot}} \makeatother   Now only one thing remains: to be consistent with text in list of tables and figures. 3 questions about Algebraic Geometry and Zariski topology I've read this exercise on my book, but I don't know how to prove it... Can someone help me?  Let X $\subseteq \mathbb{A}^n$ be an algebraic set, and $f, g:X\rightarrow k$ two regular functions.  a) If we see $f$ as an application $X \rightarrow \mathbb{A}^1(k)$, prove that $f$ is continuous in the Zariski topology.  b) If $X$ is irreducible and an open $U \subseteq X$ exists such that $f_{|U} = g_{|U}$, prove that $f=g$.  c) Give an example of $X, f$ and $g$ such that $f \neq g$ but $f_{|U} = g_{|U}$ in an open $U \subseteq X$.  Thank you!  For b) I have been able to prove that:  Let E be the subset where $f = g$ so E is closed (Why? I know that $E = (f,g)^{-1}(\Delta_{k})$ but... why is $(\Delta_{k})$ closed?). So $U \subseteq E \Rightarrow \bar{U} \subseteq \bar{E}=E$ (beacuse E is closed) $\Rightarrow X \subseteq E$, because $\bar{U}=X \Rightarrow E = X$ and $f=g$ in X. Unable to Telnet into port 25 on CentOS PostFix Server Why is my variable unaltered after I modify it inside of a function? - Asynchronous code reference