Is  k=$\sqrt{6+\sqrt{6+...}}$ monotonically increasing or is not ? i know it converges to $3$.
Let $a\ge1$ and define the sequence $(x_n)$ recursively by: $$x_1 = \sqrt{a}$$ $$x_{n+1}= \sqrt{a+x_n}$$  Here's what I did:  Plugging in some values makes it seem as if the sequence is increasing. I first solve the equation $t=\sqrt{a+t}$, equivalently $t^2 -t -a =0$, to get an idea what's going on. The roots are $$t_{1,2}= \frac{1 \pm \sqrt{1+4a} }{2}.$$ Therefore, between the roots, that is for $$t\in \left( \frac{1 - \sqrt{1+4a} }{2}, \frac{1 + \sqrt{1+4a} }{2}\right)$$  we have $t^2-t-a \le 0 \iff t \le \sqrt{a+t}$. Here's my attempt at showing inductively that $$\forall n\in \mathbb{N}, \ x_n \le \frac{1 + \sqrt{1+4a} }{2}$$  For $n=1$, the statement is true (simple arithmetic). Suppose it's true for $n$. Consider  $$x_{n+1}= \sqrt{a+x_n}\le \sqrt{a+\frac{1 + \sqrt{1+4a} }{2}}=\frac{1 + \sqrt{1+4a} }{2},$$  because $\frac{1 + \sqrt{1+4a} }{2}$ satisfies $x=\sqrt{x+a}$.  So this establishes the sequence $(x_n)$ is bounded. Now, by induction again, I try to show it's monotonically increasing. The base case $x_1\le x_2$ is obvious. Assume (strong induction) $x_{n-1}\le x_n$ for all $n&gt;1$. But since we have $$x_1\le x_n \le \frac{1 + \sqrt{1+4a} }{2}, $$   then $x_n$ lies in the appropriate interval, so $x_n \le \sqrt{a+x_n}=x_{n+1}.$  Is this proof correct? Can the condition on $a$ be relaxed to $a&gt;0$? Maybe even small negative $a$? And could someone provide me with another proof of convergence of $x_n$? I absolutely hate this one!
Was the significance of Arya's Braavosi coin ever explained? Jaqen H'ghar gives Arya the iron coin, she can apparently give it to any Braavosi and say Valar Morghulis and that gets her where she needs to go, but why?    Where do these coins come from and what do they indicate to a Braavosi? Algorithm to return all combinations of k elements from n How to change the snapping increment in different units? I would like to know how to change the increments used for snapping transforms.  This would be useful in a number of situations but my problem now is that when I change the units it does not scale the snapping increments to match.  So when holding down ctrl the cube snaps from here:    To here:    I would like it to snap to the 1 foot grid, or possibly any other amount I find useful.  It would also be nice to be able to change the rotation snap as well. How to Uninstall Blender Software I installed Blender for a project working on and it's way more than I need. I'd like to uninstall it, but can't figure out how to do so. It's not showing up in Programs (using Windows 9) and I can't find instructions on how to remove it in the Blender documentation. Any assistance would be greatly appreciated. Thanks. A ring without the Invariant Basis Number property Exponent and abelian groups What exponent $e$ can guarantee the group to be abelian? Are there any known results except the case $e = 2$? Thanks for help. Divergence of $\sum_{n=1}^{\infty} \frac{1}{2+i^n}$? I notice that,  $i^n = i, \quad if \text{ } n \equiv \text{ } 1 \text{ } mod \text{ } 4$  $i^n = -1, \quad if \text{ }n \equiv \text{ } 2 \text{ } mod \text{ } 4$  $i^n = -i, \quad if \text{ }n \equiv \text{ } 3 \text{ } mod \text{ } 4$  $i^n = 1, \quad if \text{ } n \equiv \text{ } 0 \text{ } mod \text{ } 4$  so the series $\sum_{n=1}^{\infty} \frac{1}{2+i^n}$ seems to be an alternating series that does not converge to any point in the complex plane. Is there a rigorous way to prove this divergence? How would you write a proof to this divergence or convergence (if I am wrong)?  Thanks! Insert list of figures in the table of contents Where does the word "dog" come from? Refreshing SFDC page from VF frame: CORS issue Example of a finitely generated faithful torsion module over a commutative ring Can a finitely generated module $M$ over a commutative ring have $\operatorname{Ann}(x) \neq 0$ for all $x \in M$ while $\operatorname{Ann}(M) = 0$?  It's not difficult to show that there is no such module if the ring is a integral domain. For general, I guess the answer is yes. But I failed to find a desired example. Is there a difference between foo(void) and foo() in C++ or C? Does the order of declaration matter in Java/C#? computing the orbits for a group action These are my current disk partitions. /yedek is for some personal backup files. As you can see there is only Ubuntu on my PC. I will install Windows, too. Because of this I want to shrink /dev/sda3 . It currently 546GB and I want to split it: 300GB for / and 246GB for Windows (NTFS) is OK for me.  I'm using GParted for the first time. How can I shrink /?  I don't want to lose any datas from Ubuntu. Trouble with proving $A$ is an integrally closed domain $\Rightarrow$ $A[t]$ is integrally closed domain WP List Table custom quick edit box - post meta data missing and columns change on submit I have extended class WP_List_Table to create a custom sortable table of posts along with a quick edit box with custom meta boxes.  The meta boxes populate and save but when I click submit the field values disappear and the defaults ( cb, title, tags, cats, etc) which are not even included in my class get added back to the column saved. This throws off the other columns on the page.  The saved custom field values return when the page is refreshed.  It seams like on submit the $post_object is missing.  I'm using a hacked get_inline_data function that includes the meta_values in my quick edit form instead of calling get_inline_data in my class.  My save function for the meta boxes:  add_action( 'save_post', 'save_inline_edit_meta' ); function save_inline_edit_meta( $post_id  ) {     global $pagenow;      if ( $pagenow == 'admin.php' ) {      if ( isset( $_POST[ 'is_quickedit' ] )  &amp;&amp; isset( $_POST[ 'standinghead' ] ) )         update_post_meta( $post_id, '_wnd_standing_head', $_POST[ 'standinghead' ] );      if ( isset( $_POST[ 'is_quickedit' ] )  &amp;&amp; isset( $_POST[ 'headline' ] ) )         update_post_meta( $post_id, '_wnd_alt_title', $_POST[ 'headline' ] );      if ( isset( $_POST[ 'is_quickedit' ] )  &amp;&amp; isset( $_POST[ 'deck' ] ) )         update_post_meta( $post_id, '_wnd_deck', $_POST[ 'deck' ] );     }   return $post_id;  }   My Javascript save function:  save : function(id) {         var params, fields, page = $('.post_status_page').val() || '';          if ( typeof(id) == 'object' )             id = this.getId(id);         $('table.widefat .inline-edit-save .waiting').show();         params = {             action: 'inline-save',             post_ID: id,             post_type: 'post',             edit_date: 'true',             post_status: page         };         fields = $('#edit-'+id+' :input').serialize();         params = fields + '&amp;' + $.param(params);          $.post('admin-ajax.php', params,             function(r) {                 $('table.widefat .inline-edit-save .waiting').hide();                  if (r) {                     if ( -1 != r.indexOf('&lt;tr') ) {                         $(inlineEditPost.what+id).remove();                         $('#edit-'+id).before(r).remove();                         $(inlineEditPost.what+id).hide().fadeIn();                     } else {                         r = r.replace( /&lt;.[^&lt;&gt;]*?&gt;/g, '' );                         $('#edit-'+id+' .inline-edit-save .error').html(r).show();                     }                 } else {                     $('#edit-'+id+' .inline-edit-save .error').html(inlineEditL10n.error).show();                 }             }         , 'html');         return false;     }, Why do I get 'HDIO_GET_IDENTITY failed' message when booting with external usb hard drive? I've installed Ubuntu 11.10 on an external USB (2.0) Hard Drive (/dev/sdb1) but when Ubuntu starts I get a message that says:     ata_id[279]: HDIO_GET_IDENTITY failed for '/dev/sdb': Invalid argument   where I made a mistake? The first HD contains Windows 7 (the official OS of my packard bell notebook). Using (or not) the word "to" in a sentence ending with "be able" Can words like "what" be the subject of a sentence? In a question like "Who hears a noise?",  is the subject of the sentence who?  I can think of a few tests for subjects like: "the subject is the phrase that inverts with the auxiliary to form a question". But this is a question and there is no auxiliary. We could say that's because it's a subject, but that would be just to presuppose that it's a subject in the first place.  I can think of tests like: "the form of the verb will agree with the number of the subject", but on the other hand who does not seem to have  fixed grammatical number, and it sometimes seems like who might actually be agreeing with noun phrases later in the sentence:   Who are the most prolific writers of our age? Who is the most prolific writer of our age?   There are also tests like "the subject is usually the first noun phrase in the sentence". But of course if we have a question like:   Who have you bitten?   ... you, not who seems to be the subject. The first noun phrase test is not very good.  I also know that some theories of grammar say that there is a gap in wh- questions that the wh- word is extracted from. So can this gap for instance be the real object of a question? If so, is [See community wiki post in the linked to question]  How can we show whether who is really the subject of my original example question? And how about other wh- words like what? Can they (also) function as subjects?