$$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt{5+\sqrt{ \dots }}}}}}$$   I don't understand how to solve that. I mean I don't know where to begin. Tell me if this infinite radical has a solution or converge to a number. Thanks.
How to compute $$\sqrt{(1\sqrt{(2\sqrt{(3\dots)})})}$$ &amp; $$\sqrt{(1+\sqrt{(2+\sqrt{(3+\cdots)})})}$$?  I understand that  $$\sqrt{(1\sqrt{(2\sqrt{(3\dots)})})}=(1^{1/2})(2^{1/4})(3^{1/8})\cdots$$  and  $$\sqrt{(1+\sqrt{(2+\sqrt{(3+\cdots)})})}=(1+(2+(3+(\cdots))^{1/8})^{1/4})^{1/2} $$  How to Proceed further?
heredoc for Windows batch? I have noticed that I hardly ever use the \negthinspace command  \!.  This is because I don't know how to use it properly (I do use other spacing commands quite a bit, because I can better understand the guidelines of when they might be useful). When might it be considered appropriate to use this spacing command? how to pass value like 100,102,103 to view params I am trying to zoom plot numbers.Plot numbers like 1 10-12 13/P road 20(A) 100&amp;102 and etc.I am able to zoom to all combination plot numbers except 1,2,3.when I pass such value its trowing error:  1 feature types requested, but found 2 view params specified  geoserver sql view is:  select * from plotboundary where nmindar='%nmindar%' and plno='%plno%'   default values and validation are  Name  | Default Value   | validation  nmidar| Hanagawadi I A  | ^[\w\d\s]+$  plno  |  100,102&amp;103    | ^[ A-Za-z0-9_ *@.\/#&amp;+-\,()-_]*$   ajax function:   $.ajax({             url: 'http://XX.168.1.XX:8089/geoserver/XX/ows?service=WFS&amp;version=1.0.0&amp;request=GetFeature&amp;typeName=XX:vwplot&amp;maxFeatures=5000&amp;outputFormat=text/javascript',             contentType: 'application/json',             dataType: 'jsonp',             jsonpCallback: "parseResponse",             data: {                 viewparams: "nmindar:" + nmindar + ";plno:" + plno             }   Can any one tell me how to pass value to viewparams in geoserver, value includes comma. For example as mentioned above "100,102&amp;103" is a single value(plot number).Here   comma is not a separator. Map not loading when using getLegend and showing error for getView() I was able to run my application on using Legend earlier but now I want to update my Legend when scrolling on the map. So I tried to used this code  . And I got the gist of how he is able to do that but when I am trying the same I keeping getting the error var graphicUrl = wmsSource.getLegendUrl(currentResolution);  I think this is mainly due to the wmsSource not declare. But I tried to initialize  seems wmsSource in TileWMS but does not seems to work. var getExtent = function () {     return map.getView().calculateExtent(); } var getProjectionCode = function () {     return map.getView().getProjection().getCode(); }  var updateLegend = function() {     var currentResolution = map.getView().getResolution();     var graphicUrl = wmsSource.getLegendUrl(currentResolution);        const crs = `&amp;CRS=${getProjectionCode()}`;     const bbox = `&amp;BBOX=${getExtent().join(',')}`;     const height = getHeight(getExtent());     const width = getWidth(getExtent());     const heightAndWidth = `&amp;srcwidth=${height}&amp;srcheight=${width}` ;     const legenOptions = &quot;&amp;legend_options=countMatched:true;fontAntiAliasing:true;hideEmptyRules:true;forceLabels:on&quot; ;        var img = document.getElementById('legend');     img.src = `${graphicUrl}${crs}${bbox}${heightAndWidth}${legenOptions}`; }; //Updated Legend End   //View var myview = new ol.View({     center : [8717490.201868,2788422.791843],     zoom : 4 })  var firstSel = document.getElementById(&quot;City&quot;); var secSel = document.getElementById(&quot;Sector&quot;);  //var LayerArray = [baseLayer, firstLayer]  //Map var map = new ol.Map({       target : 'mymap',       view: myview,       layers: [         new ol.layer.Tile({             source : new ol.source.OSM({                 attributions: 'Surveyor Allpication'             })         }),         new ol.layer.Tile({             source : new ol.source.TileWMS({                url: 'http://localhost:8089/geoserver/India/wms',                params: {'LAYERS': 'India:India', 'TILED': true,'TRANSPARENT':true},                ratio: 1,                serverType: 'geoserver',                transition: 0,             })         })       ] }) updateLegend();   This here is my JS &lt;div class=&quot;innerMap&quot;&gt;     &lt;div class=&quot;map&quot; id=&quot;mymap&quot;&gt;&lt;/div&gt;         Legend:     &lt;div&gt;&lt;img id=&quot;legend&quot;/&gt;&lt;/div&gt;     &lt;!--    &lt;div&gt;&lt;img src=&quot;http://localhost:8089/geoserver/wms?REQUEST=GetLegendGraphic&amp;VERSION=1.0.0&amp;FORMAT=image/png&amp;WIDTH=20&amp;HEIGHT=20&amp;LAYER=India%3AIndia&amp;RULE=Agra&quot; /&gt;%Name&lt;/div&gt; --&gt; &lt;!--    &lt;div&gt;&lt;img src=&quot;http://localhost:8089/geoserver/India/wms?SERVICE=WMS&amp;VERSION=1.1.1&amp;REQUEST=GetLegendGraphic&amp;FORMAT=image/png&amp;WIDTH=20&amp;HEIGHT=20&amp;LAYER=India%3AIndia&amp;SRS=EPSG%3A4326&amp;STYLES=&amp;SRCWIDTH=742&amp;SRCHEIGHT=768&amp;BBOX=75.4541015625%2C14.21630859375%2C91.7578125%2C31.09130859375&amp;LEGEND_OPTIONS=countMatched:true;fontAntiAliasing:true;hideEmptyRules:true&quot; /&gt; &lt;/div&gt;  --&gt;     &lt;/div&gt;  This is my HTML. How do I make this work? Difference estimate & confidence intervals for $\chi^2$ test between 2 proportions I am using a chi-sq test between two proportions (). I am using this over the z-test for proportions because I do not think my data is normal.   My question is how do I get the difference (shift) estimate &amp; confidence intervals for this test.   For example,     Data1: 193/252=.77   Data2: 154/227=.68   P-value=.032 (I used the formula in the link above. I will also add my exact code below)  Shift Estimate: I assume its the difference between proportions, ie .77-.68=.09. Is this correct?   Confidence Interval: ??????????????????  Code for generating P-value (matlab):   % Pooled estimate of proportion  p0 = (n1+n2) / (N1+N2); % Expected counts under H0 (null hypothesis)  n10 = N1 * p0;  n20 = N2 * p0; % Chi-square test, by hand  observed = [n1 N1-n1 n2 N2-n2];  expected = [n10 N1-n10 n20 N2-n20];  chi2stat = sum((observed-expected).^2 ./ expected);  p = 1 - chi2cdf(chi2stat,1);  H=0; if(p&lt;.05), H=1; end I wanted to know how I could configure multi-touch gestures in Ubuntu. In windows I can do this using the synaptics driver software.  I have already used synclient to configure the taps, palmdetect, scroll, etc. But I'd also want gesture support. Is is even possible?  System-Info: Ubuntu 12.10, Synaptics clickpad Suppose that $S_0$ and A are positive numbers, let $$S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right)$$ with $n \geq 0 $.      (a)Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$       (b)Show that $S_{n+1} \leq S_n $ , if $n \geq 1$      (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists      (d) find s   (a) Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$  Given $$P_n:  S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \geq \sqrt{A}$$  $$P_0: S_{1} = \frac{1}{2}\left(S_0 +\frac{A}{S_0}\right) \geq \sqrt{A}  $$  We assume that $P_n$ is true  $$P_{n+1}: S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right)$$ by assumption $$S_{n+2}= \frac{1}{2}\left(S_{n+1}\left(1 +\frac{A}{(S_{n+1})^2}\right)\right) \geq \frac{1}{2}\left(\sqrt{A}\left(1 +\frac{A}{(\sqrt{A})^2}\right)\right)$$ $$ S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right) \geq \sqrt{A}  $$  It follows that $S_{n+1} \geq \sqrt{A} $    (b) Show that $S_{n+1} \leq S_n $ , if $n \geq 1$  $$S_{n+1} \leq S_n$$ $$\frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \leq S_n $$ Dividing by $S_n$ $$\frac{1}{2}\left(1 +\frac{A}{S_n^2}\right) \leq 1 $$ $$\frac{A}{2S_n^2} \leq \frac{1}{2}$$ $$A \leq S_n^2$$ $$S_n \geq \sqrt{A}$$ As  $S_{n+1} \leq S_n$ yields a true statement, it follows $S_{n+1} \leq S_n$ is true.    (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists  Since $S_{n+1} \leq S_n$, the sequence is non-increasing,   using the non-increasing theorem stating that      if $\{S_n\}$ is non-increasing then $$\lim\limits_{n \rightarrow &gt; \infty} S_n = \inf\{S_n\} $$     (d) find s    Is the argumentation in (a) and (b) appropriate? Also, I have to admit I m getting less confident in my argumentation (c) and (d). How to proceed in (c) and (d)?  Much appreciated for your input or help. Imagine a black hole that is fast-approaching its final exponential throws of Hawking evaporation. Presumably, at all points in this end process there will remain a region that identifiably remains &quot;the black hole&quot; until the the very end, as opposed to huge swarm of fundamental particles that is being radiated out from it. As the mass of the black hole descends to that of individual particles, it would seem entirely feasible that the very last fermionic Hawking radiation event available to the almost-deceased black hole could leave it with an unbalanced charge, e.g. -1, and an unbalanced spin, say 1/2. It would also have some kind of mass of course, but that aspect of the final residue could be fine-tuned to any specific value by photon emissions of arbitrary frequencies. After photon emission mass trimming, the resulting black hole residuum would reach a point where it is no longer be able to evaporate into any known particle, because there is no longer any lower-mass option available to it for removing the -1 charge and 1/2 spin. The black hole residuum will at that point be stuck, so to speak, stuck with exact charge, spin, and mass features of an electron. And so my question: Is it an electron? And if so, by equivalence, is every electrons in the universe really just a particular type of black hole that cannot evaporate any further due to the constraints of charge and spin conservation? And if so, why are charge and spin so uniquely combined in such black hole remnants, so that e.g. a remnant of -1 charge and zero spin is not permitted, at least not commonly, and the mass is forced to a very specific associated level? Is there anything in the current understanding of general relativity that would explain such a curious set of restrictions on evaporation? The full generalization of this idea would of course be that all forms of black hole evaporation are ultimately constrained in ways that correspond exactly to the Standard Model, with free fundamental particles like electrons being the only stable end states of the evaporation process. The proton would be a fascinating example of an evaporation that remains incomplete in a more profound way, with the three quarks remaining incapable of isolated existence within spacetime. The strong force, from that perspective, would in some odd sense have to be a curious unbalanced remnant of those same deeper constraints on the overall gravitational evaporation process. This may all be tautological, too! That is, since Hawking radiation is guided by the particles possible, the constraints I just mentioned may be built-in and thus entirely trivial in nature. However, something deeper in the way they work together would seem... plausible, at least? If an electron is an unbalanced black hole, then the particles given off would also be black holes, and the overall process would be not one of just particle emission, but of how black holes split at low masses. Splitting with constraints imposed by the structure of spacetime itself would be a rather different way of looking at black hole evaporation, I suspect. (final note: This is just a passing thought that I've mulled over now and then through the years. Asking it was inspired by  by Ben Crowell. I should add that I doubt very seriously that that my wild speculations above have anything to do with Wheeler's concept of geons, though.) How to make an ImageView with rounded corners? On the functional square root of $x^2+1$ There are some math quizzes like: find a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$  such that $\phi(\phi(x)) = f(x) \equiv x^2 + 1.$  If such $\phi$ exists (it does in this example), $\phi$ can be viewed as a "square root" of $f$ in the sense of function composition because $\phi\circ\phi = f$.  Is there a general theory on the mathematical properties of this kind of square roots? (For instance, for what $f$ will a real analytic $\phi$ exist?) How to correctly add a path to PATH? I'm wondering where a new path has to be added to the PATH environment variable. I know this can be accomplished by editing .bashrc (for example), but it's not clear how to do this.  This way:  export PATH=~/opt/bin:$PATH   or this?    export PATH=$PATH:~/opt/bin Is it possible to apply modifiers to multiple objects at once? How is the recommended next badge determined exactly? According to , the     "Progress to next badge" bar automatically selects the badge you're closest to earning (or that's most logical to go for next)   While "the badge you're closest to earning" is pretty easy to determine: it's the bar with the most progress, I'm wondering what "most logical to go for next" means. Because right now on Meta Stack Exchange, I can track two badges: Illuminator and Legendary. My progress for Illuminator is 66.40% and for Legendary it's 66.67%, yet Illuminator is shown as the recommended next badge:    On some child metas where I barely have any activity, Student is shown as recommended next badge with a progress of 0/1, but I do have cast some votes so I have a small progress towards e.g. Civic Duty. So obviously there is some other logic at work, but what? Can I possibly recover the Windows system partition that I deleted (100MB)? After moving to SSD I deleted the System Partition (100MB) from my previous drive, causing mass destruction.  I tried to delete the System Partition which windows 7 creates (on my previous disc as I moved to SSD) and when I tried to extend the main partition size to include the new Unallocated Space it asked me if I wanted to convert the disc to Dynamic.  I simply pressed yes and the partition simply disappeared into unallocated space. All of my family pictures / videos lost.  After reading an article on Microsoft they said to recreate the partition with the same exact size, so I tried to do that but it didn't work. I did not format the drive at all when I created partitions.  How can I possibly recover the partitions?  Thanks! Proof: For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$ I need help proving the following statement:  For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$  The statement is true, I just need to know the thought process, or a lead in the right direction. I think I might have to use a contradiction, but I don't know where to begin.  Any help would be much appreciated. When can we speak of collinearity MySQL Replication : 1 Slave / Multiple Masters Disk full, du tells different. How to further investigate? I have a SCSI disk in a server (hardware Raid 1), 32G, ext3 filesytem. df tells me that the disk is 100% full. If I delete 1G this is correctly shown.  However, if I run a du -h -x / then du tells me that only 12G are used (I use -x because of some Samba mounts).  So my question is not about subtle differences between the du and df commands but about how I can find out what causes this huge difference?  I rebooted the machine for a fsck that went w/out errors. Should I run badblocks? lsof shows me no open deleted files, lost+found is empty and there is no obvious warn/err/fail statement in the messages file.  Feel free to ask for further details of the setup. Limiting range of value types in C++ Suppose I have a LimitedValue class which holds a value, and is parameterized on int types 'min' and 'max'.  You'd use it as a container for holding values which can only be in a certain range.  You could use it such:  LimitedValue&lt; float, 0, 360 &gt; someAngle( 45.0 ); someTrigFunction( someAngle );   so that 'someTrigFunction' knows that it is guaranteed to be supplied a valid input (The constructor would throw an exception if the parameter is invalid).  Copy-construction and assignment are limited to exactly equal types, though.  I'd like to be able to do:  LimitedValue&lt; float, 0, 90 &gt; smallAngle( 45.0 ); LimitedValue&lt; float, 0, 360 &gt; anyAngle( smallAngle );   and have that operation checked at compile-time, so this next example gives an error:  LimitedValue&lt; float, -90, 0 &gt; negativeAngle( -45.0 ); LimitedValue&lt; float, 0, 360 &gt; postiveAngle( negativeAngle ); // ERROR!   Is this possible?  Is there some practical way of doing this, or any examples out there which approach this? Uninstall apps locked by phone vendor