$$1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6\cdot 9}+\frac{1\cdot 3\cdot 5\cdot 7}{3\cdot 6\cdot 9\cdot 12}+\dots$$  How do I proceed on evaluating this sum.
Prove that $$1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+.........=\sqrt{3}$$   $\bf{My\; Try::}$ Using Binomial expansion of  $$(1-x)^{-n} = 1+nx+\frac{n(n+1)}{2}x^2+\frac{n(n+1)(n+2)}{6}x^3+.......$$  So we get $$nx=\frac{1}{3}$$ and $$\frac{nx(nx+x)}{2}=\frac{1}{3}\cdot \frac{3}{6}$$  We get $$\frac{1}{3}\left(\frac{1}{3}+x\right)=\frac{1}{3}\Rightarrow x=\frac{2}{3}$$  So we get $$n=\frac{1}{2}$$  So our series sum is $$(1-x)^{-n} = \left(1-\frac{2}{3}\right)^{-\frac{1}{2}} = \sqrt{3}$$  Although I know that this is the simplest proof, can we solve it any other way someting like defining $a_{n}$ and then use Telescopic sum.  Thanks.
I've always wondered about this and never found a good solution.  But  reminded me of it.  When I have a URL on my website it can be displayed and accessed any of the following ways:  http://www.somesite.com/subdirectory http://www.somesite.com/subdirectory/ http://www.somesite.com/subdirectory/index.htm http://www.somesite.com/subdirectory/index.html http://www.somesite.com/subdirectory/index.php http://www.somesite.com/subdirectory/index.asp http://www.somesite.com/subdirectory/some-relevant-keywords http://www.somesite.com/subdirectory/some-relevant-keywords.htm http://www.somesite.com/subdirectory/index.php?page=some-relevant-keywords http://www.somesite.com/subdirectory/?page=some-relevant-keywords http://www.somesite.com/subdirectory/?page=some-relevant-keywords&amp;even=more-keywords   etc...   Now, I can understand the merits of adding keywords in the URL. Even the most basic SEO guide will mention to do just that. ... but for the sake of sanity, clarity, ease of reading, ease of use, and so on, including web compliance ...  Is it preferred to have a file-extension or not?  Really, deep down my logic tells me: yes, it should. The reason being is this stems back to the days of the past when the internet was mostly USENET, FIDONET, FTP and GOPHER.   See, if a URL has no filename, then it normally is considered a directory. This is where index.htm came about, because this by default lists the directory if no index file is found. However, soon enough, web programmers started overriding this and using index.htm to actually serve the content of that web directory as a page. The main difference, was markup language was added in, and this was parsed in the browser. With this markup language, the Content-Type:text/html; tag in the response header became the indicator to what filetype it was for any file. HTML seems to be the only "filetype" that just doesn't have consistently named extensions, except for when they are saved.  Unfortunately, once web pages became the main thing, it became a security error to actually display the directory contents, so everything stayed hidden with only the actual URL content being displayed.   Not to mention the cross-platform file-naming wars.. windows based require a 3 or less digit extension, and unix/mac can have more. So should it be .HTM or .HTML or NONE and let the platform decide?  So in essence, I guess what I am trying to figure out is beyond SEO and dealing more with aesthetics and web compliance. Given Ohm's law, how can current increase if voltage increases, given fixed resistance? According to Ohm's law, $V=IR$ (voltage equals current times resistance).  So if the voltage increases, then the current increases provided that the resistance remains constant.  I know that Voltage or potential difference means work done per unit positive charge in bringing that charge from one point to another.  So according to Ohm's law, if the work done per unit charge increases then current will increase. How can this be true? Point out my mistakes. Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law] I'm trying to prove that $(ma, mb) = $|$m$|$(a, b)$ , where $(ma, mb)$ is the greatest common divisor between $ma$ and $mb$. My thoughts: If $(ma, mb) = d$ , then $d$|$ma$ and $d$|$mb$ → $d$|$max + mby$ → $d$|$m(ax+by)$. This implies that $d$|$m$ or $d$|$(ax+by)$. This is the same as  $d$|$m$ or $d$|$a$ and $d$|$b$, so $d$|$m$ or $d$|$(a,b)$. This is the same as $d$|$m|$ or $d|(a,b)$, so $d$|$|m|(a,b)$. I don't know what to do. What does the "Current event" post notice mean? The recently introduced &quot;Current event&quot;  feels incomplete:  At the very least, it needs a period at the end to complete the sentence. Aside from that, what is a future visitor of a question supposed to take away from this notice? The &quot;citation needed&quot; text explains what expectations are placed on the question bearing the notice and invites action. Is the &quot;current event&quot; notice purely informational? What's the desired effect of adding it to a question? The current on-question notice is:  This post relates to a rapidly changing event.  In addition to answering those questions, I'm hoping we can work out a proposal for a wording change to make it more clear. I can't come up with anything to start with, since I don't quite get what the notice is supposed to achieve in the first place. Similar matrices and field extensions Given a field $F$ and a subfield $K$ of $F$. Let $A$, $B$ be $n\times n$ matrices such that all the entries of $A$ and $B$ are in $K$. Is it true that if $A$ is similar to $B$ in $F^{n\times n}$ then they are similar in $K^{n\times n}$?   Any help ... thanks! I realize that I lack any numerical intuition for logarithms. For example, when comparing two logarithms like $\log_78$ and $\log_89$, I have to use the change-of-base formula and calculate the values with a calculator in order to determine which of the two is larger.  Can anyone demonstrate an algebraic/analytic method to find out which of the two is larger without using a calculator? How can I find the number of partitions of $n$ into exactly $k$ distinct parts, where each part is at most $M$?  The number of partitions $p_k(\leq M,n)$ of $n$ into at most $k$ parts, each of size at most $M$, is given by the generating function: $$  \binom{M+k}{k}_{x} = \prod_{j=1}^{k}\frac{1-x^{M+k-j+1}}{1-x^j}= \sum_{n=0}^{kM} p_{k}(\leq M,n) x^n $$  For the number of the partitions $p_k(\mathcal{D},n)$ of $n$ into at most $k$ parts there is the recurrence relationship: $$   p_{k}(\mathcal{D},n) = p_{k}(\mathcal{D},n-k) + p_{k-1}(\mathcal{D},n) $$  But what, if I want to count only the partitions with distinct parts and restricted number of parts and restricted part size?  Update: Now I know the generating function for the number of distinct restricted partitions $p_k(\leq M, \mathcal{D},n)$ of $n$ into exactly $k$ distinct parts, all at most $M$ is $$    \prod_{j=1}^{M} (1+xq^{j}) = \sum_{k,n=0}^{\infty}p_k(\leq M, \mathcal{D},n)x^{k}q^{n} $$ and there is also a recurrence relation $$    p_k(\leq M, \mathcal{D},n) =         p_{k-1}(\leq M-1, \mathcal{D},n-k) +         p_k(\leq M-1, \mathcal{D},n-k) $$ How can I prove this? Could you recommend a book, where I could read about this? Is speed of light and sound rational or irrational in nature? Just as circumference of circle will remain $\pi$ for unit diameter, no matter what standard unit we take, are the speeds of light and sound irrational or rational in nature ?   I'm talking about theoretical speeds and not empirical, which of course are rational numbers. Difference between "I'm home" vs "I'm at home" Some days I'm working with extremely large Excel files.  They have too many sheets that can't be shown in one sheet tab. (see image)      So, I'm wondering whether or not we can show Excel sheet on more than 1 row. Many tries with Google don't help much.    Many tries but I still can't make Excel show the sheet bar the way I want (≥ 2 rows), so I'm working with some VBA Scripts kopischke suggested.    FYI, right click on |&lt; &lt;&lt; &gt;&gt; &gt;| button on the bottom-right corner show you a list of up to 15 sheets.  For more, click on View more in the list appeared. Let $G\times K$ be a finite group. Suppoose $G\times K\cong H\times K$. Is this sufficient to imply $G\cong H$? Why can't I connect to Ubuntu One? I have purchase new laptop from Dell with ubuntu 12.04 preinstalled. while connecting to Ubuntu One following occurs and I am not able to sync with Ubuntu One.   WebClientErr u'Error downloading https://one.ubuntu.com/api/account/ - server replied:Not       Found' u'&lt;!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.0//EN"&gt;\n&lt;html&gt; &lt;head&gt;\n&lt;title&gt;404 Not Found&lt;/title&gt;\n&lt;/head&gt;&lt;body&gt;\n&lt;h1&gt; Not Found&lt;/h1&gt;\n&lt;p&gt;The requested URL /api/account/ was not found on this server. &lt;/p&gt;\n&lt;hr&gt;\n&lt;address&gt;Apache/2.2.22 (Ubuntu) Server at one.ubuntu.com Port 443 &lt;/address&gt;\n&lt;/body&gt;&lt;/html&gt;\n' Winter 18  does not get values from a parent record I am testing the new  &lt;lightning:datatable&gt;, it is really good, and is a good replacement for most of my use cases which today I use a custom list component that I have developed.  I am getting data from my server through apex method, to populate it into the &lt;lightning:datatable&gt; component. I can bind the returned list directly to the data attribute and if the column names are same as the field names, all data is bound correctly.  The problem occurs when I get data from a parent record. For example - I want the Account name of my contact.  This example can show the problem:  Component Markup    &lt;aura:handler name="init" value="{!this}" action="{!c.doInit}"/&gt;  &lt;aura:attribute name="mydata" 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="String"/&gt;  &lt;lightning:datatable data="{! v.mydata }"                      columns="{! v.mycolumns }"                      keyField="Id"                      onrowselection="{! c.getSelectedName }"                      onsort="{!c.updateColumnSorting}"                      sortedBy="{!v.sortedBy}"                      sortedDirection="{!v.sortedDirection}"                      hideCheckboxColumn="{!false}" /&gt;        Component Controller  ({     doInit: function (component, event, helper) {         component.set('v.mycolumns', [                     {label: 'Id', fieldName: 'Id', type: 'id', sortable:"true"},                     {label: 'First Name', fieldName: 'FirstName', type: 'text', sortable:"true"},                     {label: 'Last Name', fieldName: 'LastName', type: 'text', sortable:"true"},                     {label: 'First Name Hebrew', fieldName: 'First_name_Hebrew__c', type: 'text', sortable:"true"},                     {label: 'Last Name Hebrew', fieldName: 'Last_name_Hebrew__c', type: 'text', sortable:"true"},                     {label: 'Account', fieldName: 'Account.Name', type: 'text', sortable:"true"},                     {label: 'Email', fieldName: 'Email', type: 'email', sortable:"true"},                     {label: 'Phone', fieldName: 'Phone', type: 'phone', sortable:"true"}                 ]);          var action = component.get("c.getRecordsFromDb");         action.setParams({             sObjectName: "Contact",             commaDelimitedFieldNames: "Id,FirstName,LastName,First_name_Hebrew__c,Last_name_Hebrew__c,Account.Name,Email,Phone",             filter: ""         });         action.setCallback(this, function(response){             var state = response.getState();             if (component.isValid() &amp;&amp; state === "SUCCESS") {                 var data = response.getReturnValue();                  component.set("v.mydata", data);             }         });         $A.enqueueAction(action);     } })   APEX Aura Enabled Method  @AuraEnabled public static List&lt;sObject&gt; getRecordsFromDb(String sObjectName,         String commaDelimitedFieldNames,         String filter) {      system.debug('\n\n\n\ngetRecordsFromDb sObjectName: ' + sObjectName + ', commaDelimitedFieldNames: ' + commaDelimitedFieldNames + ', filter: ' + filter + '\n\n\n\n');      // create a dynamic soql to get the value of the field     String sql = 'SELECT ' + commaDelimitedFieldNames + ' FROM ' + sObjectName;      if (!String.isBlank(filter))         sql += ' WHERE ' + filter;      system.debug('\n\n\n\ngetRecordsFromDb sql: ' + sql + '\n\n\n\n');     List&lt;sObject&gt; records = Database.query(sql);     system.debug('\n\n\n\ngetRecordsFromDb: ' + records + '\n\n\n\n');      return records; }   The problem is with the Account.Name field - the &lt;lightning:datatable&gt; component does not display the column data.   I know that the data is returned ok, and I know that in the object that is returned, the Account is contained like an object (probably this is the problem):    Is there a good way to show parent record data with this component? How to prove $A+ A^T$ symmetric, $A-A^T$ skew-symmetric. I'm creating side skirts for a truck and I am duplicating the frames I want. The problem is it's making the duplicated object into another object. How do I put it into the same one? I just wrote an answer to , which was trying to find numeric strings with or without commas (e.g. 1,234,567.89). I wanted to be thorough because I've seen this problem a lot, both on SO and in real-world code.   When I finished, I found 7 other new answers waiting for me, mostly variations on each other. Which would be fine, except for one thing: They're garbage answers. Unless I'm severely mistaken, every answer but mine (so far) is deeply flawed, and I want to downvote all of them. Yeah, I'm a stickler on regex, but IMHO if your pattern lets stuff like  ...,.,,.,,..9 or even ... pass as "numeric input," then you've failed to meet even the bare minimum requirements.  Enough venting. The reason I'm whining to Meta about this is that I'm about to do something that'll REALLY look like tactical downvoting, and I'm mildly concerned about the consequences. If I answer a question and then rain downvotes on every competing answer, will I get flagged for abuse? That would certainly seem like suspicious behavior to me.  More of an opinion question: At what point does a semi-correct answer deserve a downvote? Obviously, if it were just a typo, I'd edit it myself. Where's the line between "close enough" and "this answer sucks?"  Basically, is it just me? You've seen the question; am I right that these answers deserve downvotes? Seven people thought that [0-9,.]+ was good enough to declare something a number, so why am I the only one who doesn't?    Note: I found some related questions, but nothing that seemed like a duplicate:    - More of an etiquette question, plus the voter hadn't posted his own answer.  - I guess this pretty much covers my second question.   Also, as some have noted, my answer ended up pretty long and thorough. I put in a lot of effort because, with all this talk about how the other answers sucked, I wanted to be extra sure mine didn't. I don't expect pages of detail in a normal SO answer. Resize multiple layers around their respective anchor points I have 50 or so layers (imported images as smart objects) that I would like to scale down, while they stay in place. Is there a way of doing this? I know After Effects has, is there a similar Scale property in photoshop? Change Folder Permissions I started off with changing to the folder I want to change permissions for, and that is the opt folder.  $ cd /opt/ test@testVM:/opt$   So I tried changing the permissions for this folder now using:  sudo chmod 775   And that hasn't worked. It showed this message:  Try 'chmod --help' for more information.   There is something I am forgetting or leaving out.  Please can you show me what I am doing wrong?  Thanks in advance. My wifi connection doesnt work properly; it stays connected only for around 5 minutes after I turn on the PC. I'm using Ubuntu 16.04.  I've tried   sudo service network-manager restart   which makes no difference  Same with   killall nm-applet &amp;&amp; nm-applet &amp;   After that I tried   sudo gedit /etc/pm/config.d/config   where I insert this string  SUSPEND_MODULES = "drivername logicalchipname"   But since I don't have permission to modify that file I tried to use chmod on it, which led to  but left me still in the same situation.  $ lspci -knn | grep Net -A3; rfkill list 03:00.0 Network controller [0280]: Realtek Semiconductor Co., Ltd. RTL8821AE 802.11ac PCIe Wireless Network Adapter [10ec:8821] Subsystem: XAVi Technologies Corp. RTL8821AE 802.11ac PCIe Wireless Network Adapter [1b9a:2482]  Kernel driver in use: rtl8821ae  Kernel modules: rtl8821ae  0: hci0: Bluetooth  Soft blocked: yes   Hard blocked: no 1: phy0: Wireless LAN   Soft blocked: no    Hard blocked: no   What do you suggest I do? Can anybody please help me to prove this:  Let $p$ be greater than or equal to $1$. Show that for the space $\ell_p=\{(u_n):\sum_{n=1}^\infty |u_n|^p&lt;\infty\}$ of all $p$-summable sequences (with norm $||u||_p=\sqrt[p]{\sum_{n=1}^\infty |u_n|^p}\ )$, there is an inner product $&lt;\_\,|\,\_&gt; $ s.t. $||u||^2=&lt;u\,|\,u&gt;$ if and only if $p=2$.