Find $\sum_{n=1}^\infty \frac{1}{n!(n^4+n^2+1)}$
Evaluation of $\sum_{n=0}^{\infty}\frac{1}{(n^4+n^2+1)n!}$
How come unpolarized light does not undergo destructive interference? As far as I know, unpolarized light is defined as light which has no clear axis of polarization, but electric field vectors in all directions. However, I also know that electric fields can be superpositioned. So this leads me to the conclusion that if unpolarized light has equal $\vec E$ vectors in all directions, they should cancel.   Obviously, though, this is wrong (we've got light!). What's the explanation? Determining the number of bit strings of length n that contains no adjacent 0s • $C_n =$ this number of bit strings  • A binary string with no adjacent 0s is constructed by   Adding “1” to any string w of length $n-1$ satisfying the condition, or Adding “10” to any string v of length $n-2$ satisfying the condition   I still don't find that intuitive at all. Why adding up $C_{n-1} + C_{n-2}$ results in $C_n$? I have tried my best and asked my friends for help, but they find the formula unintuitive as well. Could anyone here come up with a clear explanation? Thanks! the composition was  defined as follow:  (a,b) \in (R;S) &lt;=&gt; there is c | (a,c) \in R  and (c,b) \in S .   If our two relations R and S are two convex polygon   Is there a geometric interpretation of the composition of two convex polygon ? for example :   the green polygon was the result of composition. How can i describe geometrically the result !!! I've been using Ubuntu for a few years now, but I just installed Version 16.04, and I've been having a lot of issues. I'm not a command line expert, but I tried to install some software last night, and it is creating issues (and also not working properly).   When I run sudo apt-get -f install, I see the following message:  Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following additional packages will be installed:   lua-event lua-expat lua-filesystem lua-sec lua-socket lua5.1 prosody Suggested packages:   lua-zlib lua-dbi-postgresql lua-dbi-mysql lua-dbi-sqlite3 The following packages will be REMOVED:   jitsi-meet-tokens The following NEW packages will be installed:   lua-event lua-expat lua-filesystem lua-sec lua-socket lua5.1 prosody 0 upgraded, 7 newly installed, 1 to remove and 22 not upgraded. 3 not fully installed or removed. Need to get 0 B/413 kB of archives. After this operation, 2,296 kB of additional disk space will be used. Do you want to continue? [Y/n] Y (Reading database ... 176266 files and directories currently installed.) Removing jitsi-meet-tokens (1.0.962-1) ... dpkg: error processing package jitsi-meet-tokens (--remove):  subprocess installed post-removal script returned error exit status 10 Errors were encountered while processing:  jitsi-meet-tokens E: Sub-process /usr/bin/dpkg returned an error code (1)   Can someone help me figure out what I need to do to get this resolved? I'm fine with deleting jitsi, but I'm not sure if that will correct the problem.   Currently there is a red circle with a line through it in the upper right corner that indicates there may be installed packages with unmet dependencies. Accessing class members on a NULL pointer Sounds of the letter a How can I know, precisely, when to differentiate the sounds of the letter a, like in: apple and vault? How can I make a camera the active one? I have multiple cameras in a scene. What I want to do is to change which one is active. This way I can view the scene from different angles. How would I do this? I thought of this a while ago, but can't make up a proof or a counterexample. Does anyone know more about this?  $$\sqrt{a+b}  \le \sqrt a + \sqrt b , \forall a,b \in \mathbb R_+$$  Moreover, what happens with more variables? Say:  $$\sqrt{x_1+x_2+...+x_n} \le \sqrt x_1 +\sqrt x_2 + ... + \sqrt x_n $$  with $ x_i \in\mathbb R_+ \forall i \in \{1,2,...,n\} $  Or when $i = \mathbb N$  PS: I tried looking on the internet for this but I don't know how I would call this. Deciphering the RSA encrypted message from three different public keys I have three different 1024-bit public keys with common exponent $e$ but different moduli. A message $m$ is encrypted (without padding) using the three keys, which results in three different encrypted messages.  Given the three pairs of public keys $(N_i,e)$ and the encrypted messages $c_i=m^e\bmod N_i$ , how do I decipher the original message $m$ ? How do users navigate to Meta? How do you expect users to:   Know that a site's meta-site exists (and what it's for)? Find the hyperlink to the meta-site?   I know it exists, of course, and how to get there; I'm wondering how (or even whether) you expect new or naive users to do that.  I think there isn't a link to Meta from the main site, am I right (except one link in the "list of all 171 Stack Exchange sites" drop-down menu on the toolbar)?  Sometimes I tell people to post a question on Meta instead of on the main site -- I don't realise that they may not know that Meta exists, and that if they do (because they've been there before) they don't know how to get there (because they haven't yet investigated everything on the toolbar). How do I setup keyboard macros? In my daily life I'm always typing &lt;kbd&gt;Keyboard Character&lt;/kbd&gt; and I'd like to set up a macro in Ubuntu to automatically enter &lt;kbd&gt; &lt;/kbd&gt;.  I'd like to assign it to say Shift+Control+K or whatever keyboard control character combination isn't assigned and already used by Ubuntu.  Back in the day of TSR (Terminate and Stay Resident) MS-DOS programs this was possible. Certainly it must be in Ubuntu too?  Do I have to write a daemon to accomplish this?    I found other questions with no answers: ,   I found a similar question with an answer: . However that question stipulated the answer could not be AutoKey. Presentation $\langle x,y,z\mid xyx^{-1}y^{-2},yzy^{-1}z^{-2},zxz^{-1}x^{-2}\rangle$ of group equal to trivial group Problem: Show that the group given by the presentation $$\langle x,y,z  \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2} \rangle $$ is equivalent to the trivial group.   I have tried all sorts of manners to try to show that the relations given by the presentation above imply that $x=y=z=e$. However, I am stuck and would appreciate any hints as to how I should move forward. Search functionality in lightning:datatable? Is there a way to search across all rows/columns in a lightning:datatable to narrow down the list, just like jQuery DataTables? If so, how do implement that?  Can I somehow do it all on the client-side to make it super fast (some sort of DOM manipulation)  I'm having some issues with the lightning:datatable not updating. I have attached my code below, maybe you can help me spot the issue:  Markup:  &lt;aura:component extends="c:Base" controller="AccountBrowserForm"&gt; &lt;aura:handler name="init" value="{! this }" action="{! c.doInit }"/&gt; &lt;aura:attribute name="selectedAccountId" type="Id" access="public" /&gt; &lt;aura:attribute name="accounts" type="Object"/&gt; &lt;aura:attribute name="mycolumns" type="List"/&gt; &lt;aura:attribute name="sortedBy" type="String" /&gt; &lt;aura:attribute name="sortedDirection" type="Boolean" default="true" /&gt;  &lt;aura:registerEvent name="AccountFilterChange" type="c:AccountFilterChange" /&gt;  &lt;aura:attribute name="backingdata" type="List" /&gt; &lt;aura:attribute name="filter" type="String" /&gt; &lt;lightning:input type="text" onchange="{!c.filter}" value="{!v.filter}" label="Filter" /&gt;  &lt;lightning:datatable aura:id="lightningTable"                      data="{! v.accounts }"                      columns="{! v.mycolumns }"                      keyField="Id"                      onrowaction="{! c.handleRowAction }"                      onsort="{!c.updateColumnSorting}"                      sortedBy="{!v.sortedBy}"                      sortedDirection="{!v.sortedDirection}"                      hideCheckboxColumn="true" /&gt;     Controller:  doInit: function (component, event, helper) {     helper.callServer(component, "c.getAccounts", $A.getCallback(function (response) {         component.set("v.accounts", response);         component.set("v.backingdata", response);         console.log(component.get("v.backingdata") + " /// BACKING DATA INSIDE ")         console.log(component.get("v.accounts") + " // ACCOUTN DATA INSIDE")          component.set('v.mycolumns', [{                 type: "button",                 typeAttributes: {                     label: '',                     iconName: 'utility:add',                     name: 'selectRecord',                     title: 'selectRecord',                     disabled: false,                     value: 'edit',                     variant: {                         fieldName: 'variantValue'                     },                 }             },             {label: 'Name', fieldName: 'Name', type: 'text', sortable: true},             {label: 'Street', fieldName: 'BillingStreet', type: 'text', sortable: true},             {label: 'Postal Code', fieldName: 'BillingPostalCode', type: 'text', sortable: true},             {label: 'City', fieldName: 'BillingCity', type: 'text', sortable: true},             {label: 'Country', fieldName: 'BillingCountry', type: 'text', sortable: true},         ]);          console.log(component.get("v.backingdata") + " *** BACKING DATA OUTSIDE *** ")         console.log(component.get("v.accounts") + " *** ACCOUNT DATA OUTSIDE *** ")     })) },  filter: function (component, event, helper) {     var data = component.get("v.backingdata"),         term = component.get("v.filter"),         results = data,         regex;     try {         regex = new RegExp(term, "i");         // filter checks each row, constructs new array where function returns true         results = data.filter(row =&gt; regex.test(row.name) || regex.test(row.age.toString()));     } catch (e) {         // invalid regex, use full list     }     component.set("v.accounts", results); },   For some reason the filter function is completely unresponsive. Also, does it matter that I'm populating the component with a response from a callback function. @sfdcfox, I noticed in your example that you simple created an array with dummy data. My "myaccounts" attribute is of type object I found the following proof arguing for the irrationality of $\sqrt{2}$.     Suppose for the sake of contradiction that $\sqrt{2}$ is rational, and choose the least integer $q &gt; 0$ such that $(\sqrt{2} - 1)q$ is a nonnegative integer. Let $q'::= (\sqrt{2} - 1)q$. Clearly $0 &lt; q' &lt; q$. But an easy computation shows that $(\sqrt{2} - 1)q'$ is a nonnegative integer, contradicting the minimality of $q$.   Clearly, this is a vague proof in that it expects the reader to work out the math involved on his own. I took that as an exercise and worked out the following:  Assuming $\sqrt{2}$ to be rational, let $$\sqrt{2} = \frac{p}{q}$$ where $p$ and $q$ are least possible integers.  Clearly then, $\sqrt{2} - 1$ would also be rational, $$\sqrt{2} - 1 = \frac{p-q}{q}$$ where both $(p-q)$ and $q$ are also integers.  Multiplying both sides by $q$, we get $$(\sqrt{2} - 1)q = (p-q) = q'$$ where $q'$ is an integer. Using, $2&gt;\sqrt{2} &gt; 1$ , we get $$1&gt;\sqrt{2}-1&gt;0$$ proving that $q'$ is a non-negative integer.  As stated in the proof, computing $(\sqrt{2}-1)q'$ we get, $$(\sqrt{2}-1)q' = \frac{(p-q)^2}{q}$$which is not necessarily an integer (in fact necessarily not an integer). However, the proof claims it to be.  Am I missing something or something is wrong with how I proceeded?  Here is a link to the textbook where I found this proof: , (p 32, problem 1.14).  P.S. This is my first post here and my first time with TeX. I'd also like to know if there is anything wrong with the way my question is presented. Recently, I was doing my homework and I found out that Torque can be calculated using $\tau = rF$. This means the units of torque are Newton meters. Work &amp; Energy are also measured in Newton meters which are Joules.  However, torque isn't a measure of energy. I am really confused as why it isn't measured in Joules. How do you get JavaScript/jQuery Intellisense Working in Visual Studio 2008? Can I acquire a Windows 8.1 licence by free upgrade from 7 to 10, then downgrading So, I have Windows 7 pro, and I'm looking to upgrade so that I can use some of the apps that are only available in Windows 8+. I considered just taking advantage of the free Windows 10 upgrade, and actually installed it at one point, but then I found out that there's forced automatic updates and decided that wasn't for me(I'm a bit paranoid after Android 4.4.4 killed Flash Player without telling me. :P). So, I did some research.   From what I read, the Windows 10 upgrade "consumes" your Windows 7 license into a Windows 10 license, and so even if you revert back after upgrading then you can still change your mind later and jump back to Windows 10. () I also read that Windows 10 has "downgrade" rights to Windows 8 and Windows 7, which to my understanding meant that if you have a Windows 10 key, you can also use it to install Windows 8 or 7 if you want. ()  My plan for the process is re-upgrade to 10, pull the license key, downgrade to 8 and use the license key, then upgrade to 8.1 via Windows Store, and end up on 8.1. So my questions are: Is this possible?  Will it wipe all my data in the process?(not a dealbreaker if it does, but I'd need to know that so I can back it up) Do I need to re-upgrade to 10 in order to downgrade to 8?  Any input is definitely appreciated. Thanks! Why does Unix store timestamps in a signed integer? Why is a signed integer used to represent timestamps? There is a clearly defined start at 1970 that's represented as 0, so why would we need numbers before that? Are negative timestamps used anywhere? What steps can I take to try to recover lost or inaccessible data from any storage device?  Answers:   This applies to any computer storage device, e.g. internal/external hard drives, USB sticks, flash memory. The most important thing is to STOP using it, any type of I/O can ruin your chances of a recovery.     We have separate questions covering common problems with USB flash drives in greater detail: I read about Java's type erasure .  When does type erasure occur? At compile time or runtime? When the class is loaded? When the class is instantiated?  A lot of sites (including the official tutorial mentioned above) say type erasure occurs at compile time. If the type information is completely removed at compile time, how does the JDK check type compatibility when a method using generics is invoked with no type information or wrong type information?  Consider the following example: Say class A has a method, empty(Box&lt;?&nbsp;extends&nbsp;Number&gt;&nbsp;b). We compile A.java and get the class file A.class.  public class A {     public static void empty(Box&lt;? extends Number&gt; b) {} }     public class Box&lt;T&gt; {}   Now we create another class B which invokes the method empty with a non-parameterized argument (raw type): empty(new Box()). If we compile B.java with A.class in the classpath, javac is smart enough to raise a warning. So A.class has some type information stored in it.  public class B {     public static void invoke() {         // java: unchecked method invocation:         //  method empty in class A is applied to given types         //  required: Box&lt;? extends java.lang.Number&gt;         //  found:    Box         // java: unchecked conversion         //  required: Box&lt;? extends java.lang.Number&gt;         //  found:    Box         A.empty(new Box());     } }   My guess would be that type erasure occurs when the class is loaded, but it is just a guess. So when does it happen?