Solving $\sum_{k=1}^{n} k \Big( \frac{3}{2} \Big)^{k-1}$
Formula for calculating $\sum_{n=0}^{m}nr^n$
I want to install Open JDK or Oracle JDK to my machine but I got an error during installation. The error is:  Selecting previously unselected package oracle-java8-installer. dpkg: warning: files list file for package 'unity-scope-gourmet' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-guayadeque' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-gmusicbrowser' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-manpages' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-firefoxbookmarks' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-gdrive' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-home' missing; assuming package has no files currently installed E: Sub-process /usr/bin/dpkg returned an error code (2)   I searched on Google and try to something for install/reinstall this packages. Finally i try to remove Unity. I faced this error for my every tries. I am using Xmonad. How can I fix this issue? What size difference is there between the PSP 1000, 2000 and 3000 for applying Skin Stickers? Conditional Expectation of Composite Function Preliminaries  Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a complete probability space.  Let $D$ be a complete, separable, metrizable topological space with Borel $\sigma$-algebra $\mathcal{B}(D)$ (such as $D = \mathbb{R}^q$ with $\sigma$-algebra $\mathcal{B}(D) = \mathcal{B}(\mathbb{R}^d)$).  Let $\mathbb{R}$ be equipped with its canonical Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$.  Let $g: \Omega \times D \rightarrow \mathbb{R}$ be a bounded $(\mathcal{G} \otimes \mathcal{B}(D) ) / \mathcal{B}(\mathbb{R})$-measurable function.  Let $\Pi: \Omega \rightarrow D$ be a $\mathcal{G}/\mathcal{B}(D)$-measurable random variable.  Let $H : \Omega \rightarrow \mathbb{R}$ be a $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable random variable, defined by $$ H(\omega) := g(\omega, \Pi(\omega)).$$ Note, that, since $g$ is bounded, we have $H \in \mathcal{L}^2(\Omega, \mathcal{G}, \mathbb{P})$.  Let $j: D \rightarrow \mathcal{L}^2(\Omega, \mathcal{G}, \mathbb{P}) $ be defined by $$ j(\pi)(\omega) := g(\omega, \pi)  $$  For all $\pi \in D$, let $j(\pi)$ be independent of $\Pi$.  Question  I am interested in the conditional expectation $$\mathbb{E}[H \mid \Pi] :\Omega \rightarrow \mathbb{R}$$ of $H$ with respect to $\Pi$. More specifically, I suspect that (a $\mathbb{P}$-unique version of) this condititional expectation is given by  $$ \mathbb{E}[H \mid \Pi] (\omega) = \mathbb{E}[j(\Pi(\omega))], \quad (\dagger)  $$ whereby $\mathbb{E}[j(\Pi(\omega))]$ can of course also be written as $$\mathbb{E}[j(\Pi(\omega))] = \int_{\Omega} j(\Pi(\omega))(\tilde{\omega}) d\mathbb{P}(\tilde{\omega}) . $$  How can I prove, that $(\dagger)$ is the case? I have tried, tracking the definition of conditional expectation and using Fubini, but with little success so far.  Thanks for any advice! Rounding to nearest 100 First number needs to be rounded to nearest second number. There are many ways of doing this, but whats the best and shortest algorithm? Anyone up for a challenge :-)   1244->1200  1254->1300 123->100 178->200 1576->1600 1449->1400 123456->123500 654321->654300 23->00 83->100 Looking for title & author - short story - possibly called "Arms Race" - about rediscovery of manual arithmetic short story - possibly called "Arms Race" - about rediscovery of manual arithmetic.   The hero rediscovers manual calculation - e.g., how to do long division with pencil and paper - in a world where everyone has ben using calculators for aeons. Brought before the president to demonstrate this, the great line in the story is the president, after using his calculator to check the hero's answer, says, "My God -- he's right." A great national project to exploit this advance over the enemy is obviously necessary... What statistical test can I use to compare two ratios from two independent samples. The ratios are after to before results. I need to compare the after/before ratios for two independent models and show whether they are have significant difference or not. Please help! Prove that the graph is connected I was wondering if someone can help me understand how prove that this graph is connected.  Given a graph with n vertices, prove that if the degree of each vertex is at least $(n − 1)/2$ then the graph is connected.  So far I know that about connected graphs:  An undirected graph is called connected if there is a path between every pair of distinct vertices of the graph  The distance between two vertices in a graph is the length of the shortest path between them.  The diameter of a graph is the distance between the two vertices that are farthest apart. Given the Lie algebra, what is the systematic way to construct the matrix representation of the generators of the desired dimension? I ask this question here because it is the physicists for whom representation of groups is more important than mathematicians.   Let us, for example, take $SU(2)$ for concreteness. Starting from the generic parametrization of a $3\times 3$ unitary matrix $U$ with $\det U=1$, and using the formula of generators $$J^i=-i\Big(\frac{\partial U}{\partial \theta_i}\Big)_{\{\theta_i=0\}}$$ one can find the $3\times 3$ matrix representation of the generators.  However, I'm looking for something else.    Given the Lie algebra $[J^i, J^j]=i\epsilon^{ijk}J^k$, is there a way that one can explicitly construct (not by guess or trial) the $3\times 3$ representations of $\{J^i\}$? Will the same procedure apply to solve other Lie algebras appearing in physics such as that of $SO(3,1)$ (or $SL(2,\mathbb{C})$)? Why does find -mtime +1 only return files older than 2 days? Can virtual particle have mass? Do all virtual particle travel at light speed in a vacuum? else wouldn't that imply they should have rest mass however tiny? When they pop back out of existence do their mass disappear instantly? BTW what is the heaviest virtual particle ever found? I would like to know if there is any man page documenting the construction of the most basic script commands like if/then/else, while, for each, and all the relative switches, like -eq, -e, -ge, and so on. How should I calculate $\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+...+n^n}{n^n}$ How should I calculate the below limit  $$\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+...+n^n}{n^n}$$ I have no idea where to start from. On what criteria should I evaluate domain registrars? Though I've been a web developer for a fair amount of time, I am going for the first time to buy a few domain names.  I have looked into the domains I'm going to buy and know that they're available, and I've been looking into which sellers to use.  After doing a lot of research, the main ones I'm considering are 1&amp;1, Namecheap, and Gandi.  The problem is, when continuing to research, I'm not really sure what makes one domain seller distinct from another.  I don't need much in the way of services--definitely not hosting, since I plan to use Heroku for that.  I mainly only need the domain itself and DNS management, as well as possibly SSL certificates and WHOIS protection.  Question:  What makes one domain seller different from another?  How can I go about evaluating which one is the best for me?  Note:  This question is not which domain seller is the best, but rather, what criteria can I use to evaluate them and rank one over another.  I'm trying to find out what makes one domain seller different from another, since they all seem to be pretty similar to me right now. Filling between two listplots Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction How can I prove that     $$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$    for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.  Thanks Adding Layer using esriRequest Response in ArcGIS API for JavaScript? For various reasons I need to add layers to a map object using a response returned by an esriRequest to an ArcGIS Server. One reason being the data will change constantly and hard coding FeatureLayer URLs would prove inefficient.  I have successfully been able to return a JSON object containing attribution that appears to be the same data one would see on the layers REST node.     maps = [];  require([     "esri/map", "esri/layers/FeatureLayer",      "esri/dijit/Print", "esri/tasks/PrintTemplate", "esri/request",      "esri/config","esri/dijit/BasemapToggle",      "dojo/_base/array", "dojo/dom", "dojo/parser",  "dojo/domReady!"  ], function (Map, FeatureLayer,Print, PrintTemplate, esriRequest,      esriConfig,BasemapToggle,arrayUtils, dom, parser){  parser.parse();  console.log("Map Start");  esriConfig.defaults.io.proxyUrl = "http://........./proxy.ashx";  maps.addRespLyrs = function addRespLyrs (layer) {     if (layer.type == "Feature layer") {         console.log(layer.name);         maps.map.addLayer(layer);     } };   maps.handleMapInfo = function handleMapInfo (resp) {     console.log("handleMapInfo called");     maps.layers = resp.layers;     console.log(maps.layers);     for (i in maps.layers) {         var layer = maps.layers[i];         maps.addRespLyrs(layer);     } };  maps.layerRequest = function layerRequest () {     var mapInfo = esriRequest({         "url": "https://........./Application Name/MapServer/layers",         "content" : {"f":"json"},         "handleAs" : "json",         "callbackParamName" : "callback"     });     mapInfo.then(maps.handleMapInfo); };  maps.map = new Map("cpCentCent", {     basemap:"topo",     center: [-98.5795, 39.8282],     zoom: 5 });   console.log("Map End");   When attempting to add the data using the addLayer() method I recieve an error thrown from deep with in esri's API.      init.js:199 TypeError: a.id.match is not a function(…) "TypeError: a.id.match is not a function   Any thoughts as to what could be going wrong here or how to create a layer from my JSON? Is it better to difference a series (assuming it needs it) before using an Arima OR better to use the d parameter within Arima?  I was surprised how different the fitted values are depending on which route is taken with the same model and data. Or am I doing something incorrectly?  install.packages("forecast") library(forecast)  wineindT&lt;-window(wineind, start=c(1987,1), end=c(1994,8)) wineindT_diff &lt;-diff(wineindT)  #coefficients and other measures are similar modA&lt;-Arima(wineindT,order=c(1,1,0)) summary(modA) modB&lt;-Arima(wineindT_diff,order=c(1,0,0)) summary(modB)  #fitted values from modA A&lt;-forecast.Arima(modA,1)$fitted  #fitted from modB, setting initial value to the first value in the original series B&lt;-diffinv(forecast.Arima(modB,1)$fitted,xi=wineindT[1])   plot(A, col="red") lines(B, col="blue")   ADD:  Please note I am differencing the series once and fitting arima (1,0,0) then I am fitting arima (1,1,0) to the original series. I am (I think) reversing the differencing on the fitted values for the arima(1,0,0) on the differenced file.   I am comparing the fitted values - not the predictions.  Here is the plot (red is arima(1,1,0) and blue is the arima (1,0,0) on the differenced series after changing back to the original scale)  :    Response to Dr. Hyndman's Answer:  1) Can you illustrate in R code what I would need to do in order to get the two fitted values (and presumably forecasts) to match (allowing for small difference due to your first point in your answer) between Arima (1,1,0) and Arima(1,0,0) on the manually differenced series? I assume this has to do with the mean not being included in modA, but I am not entirely sure how to proceed.  2) Regarding your #3. I know I am missing the obvious, but are not $\hat{X}_t = X_{t-1} + \phi(X_{t-1}-X_{t-2}) $ and $\hat{Y}_t = \phi (X_{t-1}-X_{t-2})$ the same when $\hat{Y}_t$ is defined as $\hat{X}_t - X_{t-1}$? Are you saying I am "undifferencing" incorrectly? What is Renly's basis for removing Joffrey? Related question:   I get that he considers himself a better king than Stannis, but why isn't he ?  Did Renly know about Joffrey being illegitimate? I always thought he didn't and was distrustful that Cersei would accept Ned as Regent.  I think in both the books and the series, Renly finds out Joffrey is illegitimate from Stannis' letters, but rewatching that clips makes me think Renly was with Ned, Varys and LF in the plan to arrest Joffrey and Cersei.  [looks up something]  Oh also . So Renly did know? GoogleApiClient.isConnected() return always false How deterministic is floating point inaccuracy?