Let's say I buy a game. And after a few months, the bundle that contains the game (e.g. all games from specific publisher or developer or franchise) go on sale and I wish to buy that bundle. What happens to that game? is it removed from the bundle and I not pay for it? Do I get charged twice? Do I get the game as a gift?
I am debating whether or not to get the Potato Sack (yes, I realize I've been debating for several weeks now) bundle through steam.  I already own: Defense Grid: The Awakening, Super Meat Boy, and Audiosurf, all of which are part of the sack pack. What happens if I buy the Potato Sack?  Do I get 2nd copies of the three above mentioned games to gift elsewhere?  Is the price of the bundle reduced by the games I already have?  Do I get nothing special for already owning those 3 games?
Evaluate $\lim_{n \to \infty} \frac{1}{n}[(n +1)(n+2) \cdots (n+n)]^{\frac{1}{n}}$ Evaluate $$\lim_{n \to \infty} \frac{1}{n}[(n +1)(n+2) \cdots (n+n)]^{\frac{1}{n}}$$  We can write as $\displaystyle\lim_{n \to \infty} \frac{1}{n}[(n +1)(n+2) \cdots (n+n)]^{\frac{1}{n}}$ $$= \lim_{n \to \infty}\left[\frac{(n +1)(n+2) \cdots (n+n)}{n^n}\right]^{\frac{1}{n}} = \lim_{n \to \infty}\left[(1+\frac{1}{n})(1+\frac{2}{n})\cdots (1+\frac{n}{n})\right]^{\frac{1}{n}}$$  I have used this result if $\{a_n\}$ is a sequence of positive term which is convergent to $l$ and $b_n = (a_1. a_2 . \cdots a_n)^{1/n}$ ,thhen $b_n$ converges to $l$  here $a_n = 2$, but i think it is not right,how to use this result in this question. Give me hint how to proceed . Thank you The script I use is actually returning this output:  (source /Thetra/Commun/env_ora.sh &gt;/dev/null; sh aclenv/arun)  geIlgif0t1 -DVORACLE geIlgif0t1 prg_25900: ACOMP=X geAhadlAd1 -DVORACLE geAhadlAd1 prg_25948: ACOMP=A syCcgge0t1 -DVORACLE syCcgge0t1 prg_26023: ACOMP=S2 syVcalq0t1 -DVORACLE syVcalq0t1 prg_26071: ACOMP=Y syVvali0t1 -DVORACLE syVvali0t1 prg_26119: ACOMP=Y syAedit0t1 -DVORACLE syAedit0t1 prg_26167: (1312) DBMS WARNING: Column not found in table. (-11238)   RETURN (-6)     END  END     END     IF cm_para_f2('VDEF','abQtip_0n1','unicite',syAedit0t1:$v_unicite)&lt;&gt;0      OR null_convert$(syAedit0t1:$v_unicite,'R') &lt;&gt; 'F' THEN SET syAedit0t1:$v_unicite TO 'R'     SET $v_tips_mrgl TO 'S'     SET $v_rfbt_iban, $v_rfbt_intitule, $v_orga_creancier, $v_orga_tip_adr1, $v_orga_tip_adr2,         $v_orga_tip_adr3, $v_code_docu, $v_tips_centre, $v_code_etab, $v_orga_tip_emetteur,  $v_orga_emetteur, $v_rfbt_compte, $v_orga_banque, $v_rfbt_emetteur TO     SELECT rfbt_iban, rfbt_intitule, orga_creancier, orga_tip_adr1, orga_tip_adr2, orga_tip_adr3 ACOMP=N syAvali0t1 -DVORACLE syAvali0t1 prg_26215: (2751) DBMS WARNING: Column not found in table. (-11238)  + to_string_using$(str_to_val$(substr$(clip_str$(to_string$($f_imme_no)),1,4)),'&amp;&amp;&amp;&amp;')         + to_string_using$(str_to_val$(substr$(clip_str$(to_string$($f_sepa_propriete_no)),1,5)),'&amp;&amp;&amp;&amp;&amp;')     IF syAvali0t1:$v_unicite ='R' THEN  SET $v_sepa_rum3 TO $v_sepa_rum3 + substr$(to_string_using$(syAvali0t1:$v_ordr_direct,'&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;'),7,9)     ELSE ACOMP=N syAcalc0t1 -DVORACLE syAcalc0t1 prg_26263: ACOMP=N syAcalc0t2 -DVORACLE syAcalc0t2 prg_26311: ACOMP=N syAcalc0t3 -DVORACLE syAcalc0t3 prg_26359: ACOMP=N   To check for errors I use the following command:  (source env_ora.sh &gt;/dev/null; sh aclenv/arun) | egrep '^ACOMP=$'   Output:  (1312) DBMS WARNING: Column not found in table. (-11238)   RETURN (-6)     END  END     END     IF cm_para_f2('VDEF','abQtip_0n1','unicite',syAedit0t1:$v_unicite)&lt;&gt;0      OR null_convert$(syAedit0t1:$v_unicite,'R') &lt;&gt; 'F' THEN SET syAedit0t1:$v_unicite TO 'R'     SET $v_tips_mrgl TO 'S'     SET $v_rfbt_iban, $v_rfbt_intitule, $v_orga_creancier, $v_orga_tip_adr1, $v_orga_tip_adr2,         $v_orga_tip_adr3, $v_code_docu, $v_tips_centre, $v_code_etab, $v_orga_tip_emetteur,  $v_orga_emetteur, $v_rfbt_compte, $v_orga_banque, $v_rfbt_emetteur TO     SELECT rfbt_iban, rfbt_intitule, orga_creancier, orga_tip_adr1, orga_tip_adr2, orga_tip_adr3 (2751) DBMS WARNING: Column not found in table. (-11238)  + to_string_using$(str_to_val$(substr$(clip_str$(to_string$($f_imme_no)),1,4)),'&amp;&amp;&amp;&amp;')         + to_string_using$(str_to_val$(substr$(clip_str$(to_string$($f_sepa_propriete_no)),1,5)),'&amp;&amp;&amp;&amp;&amp;')     IF syAvali0t1:$v_unicite ='R' THEN  SET $v_sepa_rum3 TO $v_sepa_rum3 + substr$(to_string_using$(syAvali0t1:$v_ordr_direct,'&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;'),7,9)     ELSE   We can see that grep is returning non matching lines.  When I redirect the stderr to stdout with this command:  (source /Thetra/Commun/env_ora.sh &gt;/dev/null; sh /Thetra/Oracle/transfert/P2016.H.1/aclenv/arun) 2&gt;&amp;1 | egrep '^ACOMP=$'   As expected, the output is void.  Why grep is returning non matching lines when stderr is not redirected ? How can I change my default text editor from gedit to Emacs? Inequality of $\vert \text{tanh}(x)-x+\frac{x^3}{3}\vert\leq\frac{1}{8}$ for $x\in[-1/4,1/4]$ I have an assignment, that I cannot seem to find a starting point for. I am not looking for the specific answer, but only where to begin. Any pointers are appreciated.  I have to prove the inequality $\vert \tanh(x)-x+\frac{x^3}{3}\vert\leq\frac{1}{8}$ for $x\in[-1/4,1/4]$ where $x-\frac{x^3}{3}$ is the 3rd order Taylor expansion.  I've tried with the triangle inequalities, Lagrange remainders, and a lot of other stuff, but I am no where near a correct argumentation. Does anyone have any suggestions? Is there a schematic overview of Ubuntu's architecture? as enthusiastic, advanced Linux learner, I'd love to get an overview about Linux' architecure/structure in general. You know, like "the big picture".  I'm thinking of a large schematic graphic showing what is what, who is who, what system (e.g. X) comprises which subsystems (GDM/Gnome/Compiz) on the way from a to z, from boot to interactive desktop, including the most important background services (auth, network, cron, ...).   Maybe a bit like this:  but way more detailed. There's bootchart, which produces very comprehensive charts, but they again are too detailed and difficult to get the "big picture" from.  Is there such a thing? Possibly not for the whole System, but maybe for single subsystems?  I had trouble searching for this, because using search terms like "scheme" or "architecture" pointed to the wrong direction (a tool called "scheme" or CAD software for linux).  I appreciate any links. If there's interest in those schematic overviews and links, maybe someone could turn this post into a wiki post? How to create a circular shaped grid, similar to a manhole cover Why can the row-reduced echelon matrix $R$ only be identity matrix? I'm reading Linear Algebra by Hoffman, Kunze where the authors explained that a $n\times n$ matrix $A$ being invertible is equivalent to the fact that $A$ is row-equivalent to $n\times n$ matrix $R$ which is an identity matrix.  In the proof of the theorem, they wrote:     $$R= E_k\ldots E_2E_1 A$$ where $E_1,\ldots,E_k$ are elementary matrices. Each $E_j$ is invertible, and so $$A = E_1^{-1}\ldots E_k^{-1}~R\,. $$  ... Since, $R$ is a (square) row-reduced echelon matrix, $R$ is invertible if and only if $R=I\,.$  [...]   I couldn't get the conclusion, since any row of $R$ can't be zero, it has to be identity matrix $I\,.$ Why is it so?  Isn't there any other row-reduced echelon matrix other than the identity matrix having no zero row and invertible? Why is it so? Do Stack Exchange sites allow editing of comments and posts from other users? Possible Duplicate:         Is this normal behavior in Stack Exchange sites that after just signing up one can edit not only his/her own comments / posts, but also others' comments / posts?  If so, then the Q&amp;A does not look safe to me. Imagine someone maliciously modifying a post and bringing bad reputation on a Q&amp;A member.  Can you check this and perhaps fix it? Spacing issue for newenvironment wrapper of equation environment If I create a custom environment based on the equation environment, I get a spacing issue: the next line of text after my environment closes doesn't start on the left margin.  Example code:  \documentclass{article} \begin{document} \newenvironment{myenv}{\begin{equation}}{\end{equation}} \noindent $| \leftarrow$ this is the left margin. \begin{myenv} e ^{i \pi} + 1 = 0 \end{myenv} $| \leftarrow$ this isn't the left margin. \end{document}   Output:    Removing the newline after \end{myenv} solves the spacing problem, but the resulting code is not readable.  Of course, in this example I could just use the equation environment itself. My real issue involves a more complex custom environment that suffers from the same problem, and which also includes an equation environment within it. Turning a cipher into a hashing function I'm having trouble with this functional equation, to find all functions $f:\mathbb R \to \mathbb R$ for which the following is valid for all $x,y\in \mathbb R$: $f(x) + f(x+f(y)) = y + f(f(x) + f(f(y)))$ I haven't been able to pinpoint any solution, nor to prove surjectivity or injectivity. So 0 progress. Tried with $x=y=0, x=0, y=0, x=-f(y), y=f(x)$, and even with some assumptions on bijectivity, but nothing helped. I don't know the source but seems like a pretty difficult problem, maybe even impossible to solve. How to use a C++ string in a structure when malloc()-ing the same structure? Comparing negatives: "she seems not to know" vs. "she doesn't seem to know" What is the difference in style and meaning between the following two:        She seems not to know.   She doesn't seem to know.      Is there a name to this type of construction? I've been try to download Xcode for work, but for some reason the Mac App Store is giving me a bit of trouble. Activity monitor shows some network usage, but in Launchpad it seems the download isn't moving.   When I go to the purchases panel of the Mac App Store, I see this.     No status, no pause button, in fact - the button has no text on it! I've tried restarting the App Store, restarting my computer, even removing the com.apple.appstore folder in /var/folders but alas, no luck.  Can anyone diagnose the issue? When I removed com.apple.appstore, Xcode still showed up on Launchpad, and clicking on it did nothing - it was frozen in time. So can I kill off this partial download and start new? I have a WordPress install on a dedicated server with decent memory (upwards of 8GB).  The WordPress install includes many plugins - 50 active plugins. Most important to note is the  plugin to make WordPress multilingual. The theme is .  The stack is LAMP - CentOS 6.3, Apache 2.4.4, Percona MySQL, PHP-FPM.  I have been quite stupid, and didn't set up a test or staging server.  Everything was running fine till I updated to WordPress 3.8, and also updated several plugins at the same time.  I didn't pay attention at the time, but all my widgets got messed up, and so did any sort of navigation etc.  I enabled error_log for a bit, i.e.  ini_set('log_errors',TRUE); ini_set('error_reporting', E_ALL); ini_set('error_log', dirname(__FILE__) . '/error_log.txt');   This resulted in an immense file - basically the error_log grows by about a GB every 4 minutes or so.  The primary error I found in the error log was database server disconnect, better known as      MySQL server has gone away   The first priority was to patch the problem. So I installed  plugin. This mostly, though not completely solved the issue. Many of the widgets came back.  However, the special (non-English) characters are all messed up (converted to question marks). I tried to fix this by opening wp-mysql-ping.php and commenting out   if ( !DB_CHARSET &amp;&amp; version_compare(mysql_get_server_info($this-&gt;dbh), '4.1.0', '&gt;='))                                     {                                         $this-&gt;query("SET NAMES '" . DB_CHARSET . "'");                                     }   Didn't help.  Besides, the plugin is not a permanent or complete solution. So leaving the site in this messed up state, I started working on other possibilities:  As suggested at , I edited my.cnf to add  wait_timeout = 60   Didn't help.  Also added caching to my.cnf  query_cache_type = 1 query_cache_limit = 8M query_cache_size = 512M   Didn't help.  (and yes, I restarted mysql every time I edited my.cnf)  Also tried the method : In wp-db.php before  $this-&gt;select( $this-&gt;dbname, $this-&gt;dbh );   I added  $this-&gt;query("set session wait_timeout=600" );   Didn't help.  I'm at my wit's end. Please help! "To which", "by which", "on which" etc I have come across the phrases like "to which","for which", "by which", "on which" and so on(using a preposition with a relative pronoun).   e.g. The chair on which the body was found..  Could someone please explain if there's a particular rule to be followed when building/using them?   Can I use any preposition before 'which'?  What if I say "The chair which the body was found on"? How would I decide which preposition to be put? Can I use other relative pronouns too?  Thanks in advance. Why does hair grow after trimming but remains at a constant length after a while? Some hair especially body hair regrows after trimming but stops growing after a while. What is the mechanism behind control of hair growth and how is the length limit determined? Boundary condition with spatial derivative is ignored by NDSolve In templates, where and why do I have to put typename and template on dependent names? What exactly are dependent names anyway?    I have the following code:  template &lt;typename T, typename Tail&gt; // Tail will be a UnionNode too. struct UnionNode : public Tail {     // ...     template&lt;typename U&gt; struct inUnion {         // Q: where to add typename/template here?         typedef Tail::inUnion&lt;U&gt; dummy;      };     template&lt; &gt; struct inUnion&lt;T&gt; {     }; }; template &lt;typename T&gt; // For the last node Tn. struct UnionNode&lt;T, void&gt; {     // ...     template&lt;typename U&gt; struct inUnion {         char fail[ -2 + (sizeof(U)%2) ]; // Cannot be instantiated for any U     };     template&lt; &gt; struct inUnion&lt;T&gt; {     }; };   The problem I have is in the typedef Tail::inUnion&lt;U&gt; dummy line. I'm fairly certain that inUnion is a dependent name, and VC++ is quite right in choking on it. I also know that I should be able to add template somewhere to tell the compiler that inUnion is a template-id. But where exactly? And should it then assume that inUnion is a class template, i.e. inUnion&lt;U&gt; names a type and not a function? Responsive Web Design - what should happen when the mobile device is rotated? Currently, the standard behaviour for viewing web pages on a mobile is that:   If viewed in portrait the text is smaller because the width of the device is small and the whole page width needs to fit into view. If viewed in landscape the view 'zooms in' as the width is now wider so that the text is larger and more readable, but with less content on screen as a result.   So, this brings a dilemma. When creating a responsive website what should happen on device rotation? (Not on initial page load, I am referring to what should happen once the site is loaded and you decide to then rotate the screen)   When rotated from portrait to landscape should you:   a) Re-detect the device widths and map the contents according to the new dimensions (thereby retaining the same font / images sizes as when viewing portrait). b) Keep the same dimensions and on rotation just bring the content to fit the window (thereby zooming in the content to make it more readable for viewers if the portrait orientation text is too small).          &ndash; Wireframes created with   Bear in mind that with a traditional web-page viewed in the mobile you have the option of pinch-to-zoom the text, so if it's too small you can zoom in. With a responsive site this isn't an option because the site will fit the dimensions so there is nothing to zoom into.  (I guess there is a third option too - use responsive fonts so that the font sizes change according to the screen width. But that's probably a different question altogether).