Help trying to prove the existence of sequence limit.
Proving the general limit $\lim_{n\to\infty} \frac{an+c}{bn+d} = \frac{a}{b}$
Alternative to Adobe Illustrator Can you recommend a good alternative to Adobe Illustrator?   I am learning vector based graphic design, but I am not ready to make such a large investment as to purchase a copy of Adobe Illustrator. Also, Illustrator might be a little more power than I need at the moment. I am using a Macintosh. List of lists changes reflected across sublists unexpectedly How to set scale across all blender objects and keyframes? Visual Flow: How to set Date field to Null / Blank? Can I get from Gatwick airport to London by Oyster card? I read many contradictions as to whether one can use a Oyster card from Gatwick Airport to central London.  What is the most time-efficient way to go from Gatwick to "Old Billingsgate" (Monument tube station is nearby)? Public transportation is preferred over taxi. I am flying to Tbilisi, Georgia with Turkish Airline. My stay in Istanbul is 12 hours. Can I get transit visa to visit Istanbul city ?    I am from Pakistan with Oman employment visa. Update values in one table based on contents of another Software Name: Oracle SQL Developer Version 17.2.0.188  I have three tables: orders, order_details, and products.  I wanted to run a query to update units_in_stock in the products table based on quantity in order_details, where the associated orders row has been marked as shipped, and has a shipping date.  Here's what I tried:  UPDATE Products SET (Products.Unit_In_Stock = Products.Unit_In_Stock - Order_Details.Quantity) INNER JOIN (     Orders INNER JOIN Order_Details ON Orders.Order_ID = Order_Details.Order_ID     ) ON Products.Product_ID = Order_Details.Product_ID WHERE (         ((Orders.Shipped_Date) IS NOT NULL)         AND ((Orders.This_Order_Was_Shipped) = "YES")         );   When I run this, I get:  Error at Command Line : 1 Column : 45   Line 1, Column 45 is the equal sign in the very first statement that said  Products.Unit_In_Stock = Products.Unit_In_Stock - Order_Details.Quantity   Can you help me get this working? using System; using System.Security.Permissions; using Microsoft.SharePoint; using Microsoft.SharePoint.Security; using Microsoft.SharePoint.Utilities; using Microsoft.SharePoint.Workflow;  namespace Change_Title_Event.Change_Title {     /// &lt;summary&gt;     /// List Item Events     /// &lt;/summary&gt;     public class Change_Title : SPItemEventReceiver     {        /// &lt;summary&gt;        /// An item is being added.        /// &lt;/summary&gt;        public override void ItemAdding(SPItemEventProperties properties)        {            properties.AfterProperties["Title"] = "Kevin: Changed by event receiver at" + DateTime.Now.TimeOfDay.ToString();            base.ItemAdding(properties);        }       } }   Where simple example, which works perfect on my local Sharepoint 2010. It is sandboxed.  But when I package this solution, it do not work on Sharepoint Online. I does not give any Errors either.  I think I am doing something wrong, something simple, but I can't simply find anything on Google about it.  Here is another who asks on the MSDN forum, but gets a very wierd reply, that I do not understand. How to show that any separable space is CCC How can I draw a dot under a symbol in math mode? What's the Pythonic way to combine two sequences into a dictionary? Is there a more concise way of doing this in Python?:  def toDict(keys, values):   d = dict()   for k,v in zip(keys, values):     d[k] = v    return d Where is the rest of Yoda's species? We never seem to see any beings of Yoda's species - except, of course, Yoda himself.  This leads us to the question of where the rest of the li'l critt'rs are. Yoda had to have parents, correct? So what happened to them? Was Yoda, put simply, just one of the last of a dying species? My question was closed for being a "shopping list question", even though it was very much related to the topic of the site. Why? Can I visit the UK with an EU permanent residence card? I am non EU national, living in a Schengen country for past 5 years. I hold Indian passport and EU permanent residence card valid for 5 years.   Can i go to UK for few days to visit friends without a visa?  I am aware that UK is not in Schengen, and Schengen visas and residence permits issued by Schengen country is not valid for United Kingdom? But what about the PR card issued by a Schengen country?  I am also aware that EU citizens carrying EU passport have no restrictions to visit UK. How do I make ditto marks? In France, students are taught not to repeat the same text twice when there is little difference between a line and the one below. It's a quick way to take notes.  I reproduced this effect by tinkering:    The code is as follows:  \documentclass[aspectratio=1610]{beamer}  \usepackage[T1]{fontenc} \usepackage[french]{babel}  \usepackage{amsmath, amsfonts, amssymb} \usefonttheme[onlymath]{serif}  \begin{document} \begin{frame} Lorsque le \og \textit{carreau carré} \fg{}   mesure: \begin{itemize} \item $\rm 1\,cm$ de côté, l'unité d'aire est appelée le \og \textrm{centimètre carré} \fg{} et est noté $\rm cm^2$; \item $\rm 1\,dm$ \hspace{1mm}'' \hspace{3mm}'' \hspace{8mm}'' \hspace{8mm}'' \hspace{5mm}''\hspace{8mm}'' \hspace{5.5mm}''\hspace{1mm} \og \textrm{décimètre carré} \fg{} \hspace{2mm}'' \hspace{2mm}'' \hspace{4mm}'' \hspace{1.5mm} $\rm dm^2$; \item $\rm 1\,m$ \hspace{3mm}'' \hspace{3mm}'' \hspace{8mm}'' \hspace{8mm}'' \hspace{5mm}'' \hspace{7mm}'' \hspace{5.5mm}''\hspace{1mm} \og \textrm{mètre carré} \fg{}  \hspace{9mm}'' \hspace{2mm}'' \hspace{4mm}'' \hspace{1mm} $\rm m^2$; \item etc. \end{itemize} \end{frame} \end{document}   How can I get the same effect more simply than by tinkering like I just did? Best way to follow a log and execute a command when some text appears in the log If I want to solve the equation $4x = e^x$, is there a way to solve for $x$ without graphing and looking for intersection? Variance of Normal Order Statistics Suppose we have $X_1, \cdots, X_n \overset{\textrm{i.i.d.}}{\sim} \mathcal{N}(0, 1)$ with $n &gt; 50$, and let $X_{(1)}, \cdots, X_{(n)}$ be the associated order statistics. Are there any references pointing to formulas that specify (or estimate) the variance of the $k^{\textrm{th}}$ order statistic, i.e. $\textrm{Var}(X_{(k)})$, for $1 \leq k \leq n$? I am looking for formulas can be explicitly evaluated without resorting back to numerical integration.  I have seen this question: , which attracted a number of answers on approximate formula for the mean of the $k^{\textrm{th}}$ order statistic, e.g. Blom (1958) (and modifications based on work by Harter (1961) and Elfving (1947)). Searching on Google also yields a number of work on the variance / standard deviation for the $k^\textrm{th}$ order statistics for small $n$, all of them opting for a computation/tabulation route. They are: Godwin(1949), Teichroew (1956), and Parrish (1992). I have also attempted to derive something myself following the method sketched by @probabilityislogic in , along the lines of: $$\textrm{Var}(X_{(k)}) = \int_{-\infty}^{\infty} \left(x-\mathbb{E}(X)\right)^2 \frac{n!}{(k-1)!(n-k)!} f_X(x)[1-F_X(x)]^{n-k}[F_X(x)]^{k-1}\;\textrm{d}x,$$ with lower case $f$ denoting the PDF of a r.v. $X$, and upper case $F$ denoting the CDF. Noting $x = F^{-1}(F_X(x))$ and using the substitution $u = F_X(x)$ (with corresponding r.v. in upper case $U$), we have: $$\textrm{Var}(X_{(k)}) = \int_{0}^{1} \left(F_X^{-1}(u) - \mathbb{E}\left(F_X^{-1}(U)\right)\right)^2\,\mathcal{B}(u | k, n-k+1) \; \textrm{d}u,$$ where $\mathcal{B}$ denotes the PDF of a beta distribution (not the beta function). This can be rewritten as the variance based on the beta distribution: $$\textrm{Var}(X_{(k)}) = \textrm{Var}_{\mathcal{B}(u | k, n-k+1)}\left(F_X^{-1}(u)\right).$$ Approximating the RHS using (the variance bit of the) delta method, we get: $$\textrm{Var}_{\mathcal{B}(u | k, n-k+1)}\left(F_X^{-1}(u)\right) \approx \textrm{Var}_{\mathcal{B}(u | k, n-k+1)}(U) \cdot \left[(F_X^{-1})'\left(\mathbb{E}_{\mathcal{B}(u | k, n-k+1)}(U)\right)\right]^2,$$ where the prime denotes the derivative. The first term of the product is standard result. For the second part, given $(F^{-1})'(\cdot)=\frac{1}{f(F^{-1})(\cdot)}$ (see e.g. ), we then have: $$\textrm{Var}_{\mathcal{B}(u | k, n-k+1)}\left(F_X^{-1}(u)\right) \approx \frac{k(n-k+1)}{(n+1)^2(n+2)} \frac{1}{\left(f\left(F^{-1}\left(\frac{k}{n+1}\right)\right)\right)^2}.$$ Substituting in the standard normal distribution ($\phi$ for PDF, $\Phi$ for CDF) and scaling it to the desired variance $\sigma^2$ we arrive at: $$\textrm{Var}(X_{(k)}) \approx \frac{k(n-k+1)}{(n+1)^2(n+2)} \frac{1}{\left(\phi\left(\Phi^{-1}\left(\frac{k}{n+1}\right)\right)\right)^2},$$ which we can throw in a $\sigma^2$ term in front if it is not a standard normal. But surely someone must have derived that somewhere (again I am looking for some references), and there is a high chance that I have gone off track somewhere above. a, b are integers. p is prime. I want to prove: $(a+b)^{p} \equiv a^p + b^p \pmod p$  I know about Fermat's little theorem, but I still can't get it I know this is valid: $(a+b)^{p} \equiv a+b \pmod p$ but from there I don't know what to do.  Also I thought about $(a+b)^{p} = \sum_{k=0}^{p}\binom{p}{k}a^{k}b^{p-k}=\binom{p}{0}b^{p}+\sum_{k=1}^{p-1} \binom{p}{k} a^{k}b^{p-k}+\binom{p}{p}a^{p}=b^{p}+\sum_{k=1}^{p-1}\binom{p}{k}a^{k}b^{p-k}+a^{p}$ Any ideas?  Thanks! When to use static methods