How do I prove that for all positive integer n, the inequality $2n\choose n$$&lt;4^n$ holds?  Thank you!
I would like to prove the following inequality, for $n=0,1,2,...$,   $$ \binom{2n}{n}\leq 4^n.$$ I already proved it by induction, and I'm looking for another proof.
Connection pools being reset with Error: 18056, Severity: 20, State: 46. & Perfmon Counters not showing We are using SQL Authentication (to reduce the number of connection pools) &amp; .NET 4.0 connection strings to connect to SQL Server Enterprise Edition 2012 SP1 on a Windows 2008 R2 Enterprise Server:     Microsoft SQL Server 2012 (SP1) - 11.0.3000.0 (X64)   Oct 19 2012 13:38:57   Copyright (c) Microsoft Corporation   Enterprise Edition (64-bit) on Windows NT 6.1  (Build 7601: Service Pack 1)   We use about 50 servers split into 8 different groups different parts of a website.  Our website is using this SQL Server to log visit tracking data. Over the last few days it has spat out the following messages about the resetting connection pools:     The client was unable to reuse a session with SPID 1327, which had been reset for connection pooling. The failure ID is 46. This error may have been caused by an earlier operation failing. Check the error logs for failed operations immediately before this error message.   Error log reads:     Error: 18056, Severity: 20, State: 46.   The client was unable to reuse a session with SPID 959, which had been reset for connection pooling. The failure ID is 46. This error may have been caused by an earlier  operation failing. Check the error logs for failed operations immediately before this error message.   Login failed for user 'xxxx'. Reason: Failed to open the database 'xxxxxxxx' configured in the login object while revalidating the login on the connection. [CLIENT: 10.xx.xx.xxx]   After some digging, I found this document on the CSS blog:  and this one by Aaron Bertrand: . I know the error number is different but the failure ID is the same with a number of the messages are identical).  Failure ID 46 suggests that the login did not have permissions. Our logins default to the master database and the db name is specificied in the connection string.  I wanted to check the number of connection strings pools, etc. and checked all the counters in Perfmon for .Net Data Provider for SqlServer. It only gave me the option of defaultdomain9675 for the instance, so I selected that assuming that is a system generated ID name for our Datacentre network. Unfortunately all of the counters are reading zero. On one of our other main servers the connection pools are hovering around 10 which is what I expected to see on a healthy server with that kind of load.  My question is 3 fold   Can anyone suggest why the Windows 2008 R2 Server is not showing .Net Data Provider for SqlServer? Has anyone experienced this as I obviously believe that the login not having permissions is a red herring ? If different groups of web servers have the same connection string syntax but with slightly different whitespace, would this cause the server to use another connection pool?   Min &amp; max memory settings are 20GB &amp; 58GB respectively. The server is a dedicated database server with 64GB of RAM. I don't think memory is the issue as the box seems to have a decent page lfe expectancy. Auto close is not enabled. The server is always up: this is a 24x7 website with heavy usage. Alt-Tab shows two application switchers on Unity I started to have problems with my Ubuntu application switcher (Alt +  Tab) this morning. When I press Alt +  Tab, not only one, but two application switchers show up. Last night I customized some stuff using the Unity Tweak Tool. I thought I didn't touch the application switcher. So now it looks like this: How did Sauron remain in control of the Nazgûl? Combine 4 cells in a table Here is a simple table.  I want to merge 4 cells. I want to combine the three "this" cells area in this table. I thought I have to use multirow and multicolum at same time. However, I could not find how to use it.  If possible, could you post the modified LaTeX source code?     My LaTeX code:  \documentclass{article} \usepackage{multirow}% http://ctan.org/pkg/multirow \usepackage{hhline}% http://ctan.org/pkg/hhline \begin{document}  \begin{table}[ph] \begin{center} \begin{tabular}{c|c|c|c|c} \hline \multirow{2}{*}{Raaa (k)} &amp; \multicolumn{4}{c}{C ()} \\ \hhline{~----}  &amp; 3.3 &amp; 2.5 &amp; 1 &amp; 0.5 \\ \hline \multirow{2}{*}{Raaa (k)} &amp; \multirow{2}{*}{this} &amp; this &amp; 0.5 &amp; 0.6\\ \hhline{~~~--}            &amp;                       &amp; this &amp; 0.7 &amp; 1.2 \\ \hline \end{tabular} \caption{R, C ripple size} \label{T:peak} \end{center} \end{table} \end{document} When speaking precisely or technically, one would say that "Homo erectus and homo sapiens are two species of hominid" rather than "Homo erectus and homo sapiens are two species of hominids." The hominid here should be singular because we are speaking about instances of a single class ("class" being used here in its broader sense, not in the sense of taxonomic grouping).  Now let's consider more common parallel constructions replacing the word species with other words.      Tide and Wisk are two brands of detergent.       Tide and Wisk are two brands of detergents.      Cats and dogs are two types of pet you can buy at Pets-R-Us.       Cats and dogs are two types of pets you can buy at Pets-R-Us.   I feel that using the singular noun to specify the class is more grammatically correct, but sounds stilted in conversation. Almost everyone I hear making these constructions uses the plural form. What are your thoughts about singular vs. plural here?  Edit  Here's a more extreme example that may help clarify the issue. Consider the following sentence:     It was my first attempt at calming an angry crowd, and I just        stood there while people were hurling all kinds of insults at me.   Here I think the singular insult sounds strange, if not downright wrong.  Further Edit  I just noticed in A Treasury for Word Lovers (Morton S. Freeman, 1983) a section entitled "Types of Errors." This is a book about (American) English usage by an English professor and editor, which purports to be a "practical guide for serious writers and readers." I wish I could say this discovery satisfies the question, but in fact I now feel somewhat farther from the truth, if there is any single "truth" in this matter. But at least I don't have to worry about being wrong when using the plural form of the class in these constructions. Is there a word for capitalizing in the middle of a word, as in "eBay"? Usually this sort of word is a trademarked name, such as eBay or BlackBerry.  Is there a word that describes capitalization mid-word? How to set a font for math mode? I would like to have something like Segoe UI of something similar. Thank you  \documentclass{article} \usepackage{fontspec} \usepackage{firamath-otf} \begin{document} \begin{equation} x+y=45 \end{equation}  \end{document}   Error: Finding the maximum value of $\int_0^1 f^3(x)dx$ Find the maximum value of $\int_0^1 f^3(x)dx$ given that $-1 \le f(x) \le 1$ and $\int_0^1 f(x)dx = 0$   I could not find a way to solve this problem. I tried to use the cauchy-schwarz inequality but could not proceed further  $$\int_0^1 f(x) \cdot f^2(x) dx \le \sqrt{\left(\int_0^1f^4(x)dx\right) \left( \int_0^1 f^2(x) dx\right)}$$  Any hints/solutions are appreciated. Persistent Network Configuration using KVM and libvirt Why do Lagrange Multipliers work? I know that the Lagrange multiplier method helps us evaluate critical points of $f$ on the closed boundary of the restriction.   In other words we solve:$$\nabla f=\lambda \nabla g$$  But why does this method actually work? Can someone please give me an explanation. Thank you! Routing sound from different cards to a single output I have a setup where I am monitoring for silence on an incoming stream on 3 different audio cards.  When I detect silence on the master I switch to the secondary, and if I detect silence there I switch to the last card.  The way I have it set up I load/unload the loopback module for each card depending on who's the active card.  This has the inconvenience that I need a 3-to-1 minijack cable to be able to maintain the stream.  Can I load the loopback module and point the output to a single card (independently of the input card)?  How can I route the audio from any input to a single output? How can I make sure that only low and medium security inmates are in a specific area? Is it possible to reverse the overlay you can set? It seems like you are only able to clear areas for low OR medium security prisoners, but sharing is not possible. The thing I want here is to make sure that max security prisoners are not in certain areas. Why are map tiles being repeated? Using MapServer and Openlayers Problem I have: map tiles from my layer are being repeated.  Shapefile is projection 3059. I have specified 3059 is in the mapfile and also when creating map object.  I know OpenLayers.Layer.WMS is recommended, but when I use it I only see empty map. When I use OpenLayers.Layer.MapServer, map is displayed (and tiles are repeated), so I'm using that for now.  My layer (EPSG:3059) looks like this    Versions: Openlayers 2.13.1 MapServer 6.0.3  WMS Server mapfile:  MAP     NAME country_map     STATUS ON     SIZE 800 500     EXTENT 313187.4 146161.56 766469.92 470367.64     UNITS METERS     SHAPEPATH "shapefiles"     IMAGECOLOR 155 155 155      WEB         TEMPLATE "template.html"         IMAGEPATH "C:/MS4W/ms4w/apache/htdocs/gis/output/"         IMAGEURL "/gis/output/"          METADATA             "wms_title"             "MapServer + Openlayers"             "wms_onlineresource"    "http://localhost:1025/cgi-bin/mapserv.exe?map=C:\MS4W\ms4w\Apache\htdocs\gis\mapfile2.map"             "wms_enable_request"    "*"             "wms_srs"               "EPSG:3059"         END     END      PROJECTION         "init=epsg:3059"     END  # example layer     LAYER     METADATA         "wms_title"             "lat_rob_region"     END     NAME "lat_rob_region"     DATA lat_rob_region     STATUS ON     TYPE POLYGON         CLASS             NAME "lat_rob_region"             STYLE                 COLOR 0 255 0             END         END     END END   Javascript:  var lon = 25; var lat = 57; var zoom = 5; var map, layer1;  function init() {     map = new OpenLayers.Map( 'map',{displayProjection: new OpenLayers.Projection("EPSG:3059")} );      layer1 = new OpenLayers.Layer.MapServer( "Country map",              "http://localhost:1025/cgi-bin/mapserv.exe?map=C:/MS4W/ms4w/Apache/htdocs/gis/mapfile2.map", {layers: 'lat_rob_region'},             {gutter: 15},{srs: 'EPSG:3059'});     map.addLayer(layer1);      map.setCenter(new OpenLayers.LonLat(lon, lat), zoom);     map.addControl( new OpenLayers.Control.LayerSwitcher() ); } Is it alright to use lowercase "i" or should you always use "I" (uppercase)? I already read it from manual but I can't see difference..  su - change user ID or become superuser  sudo -s [command]   The -s (shell) option runs the shell specified by the SHELL environment variable if it is set or the shell as specified in passwd(5).  If a                    command is specified, it is passed to the shell for execution.  Otherwise, an interactive shell is executed.  sudo -i disappear description in manual I'm looking for a way to get the Default currency symbol (in a none-multi-currency org) of the salesforce org locale (to have it displayed on a visualforce page in a field that can't be &lt;apex:inputField&gt; or &lt;apex:outputField&gt;) preferably query-able from apex (via the database, a describe or whatever) and then again the actual symbol, meaning € for Euro, $ for dollar, and so on...  There must surely be some sort of mapping hidden in the code somewhere salesforce does this via the input/outputfields if bound to an sobject of type currency  Can anyone point me in the right direction? A simple stupid question that I should already know the answer to:  Is this true: $\left|z^5\right| = \left|z\right|^5$ where z is a complex number with a non-zero imaginary component.  Thanks in advance. Automatically redirect anonymous user from unanswered duplicate question to corresponding answered version Duplicate questions that use different words to describe the same problem are important for people to find the content via search engines. But the user experience when an anonymous user arrives at a duplicate is less than ideal.  They have to notice the duplicate link and understand that the answer they are looking for is likely behind that link. I don't think this is all that obvious to someone without experience with an SE site.  I propose to just redirect users to the duplicate automatically if the closed duplicate has zero answers. In that case they need to follow the link to the duplicate in any case, so we might as well redirect them to it in the first place.  I could imagine this being problematic with search engines, as we would be displaying different content than what the search engines see. If that is too problematic I would suggest to show a very obvious link to the duplicate in the answers section for anonymous users. Show that one-sided limits always exist for a monotone function (on an interval) Show that one-sided limits always exist for a monotone function on an interval $[a,b]$.  Me attempt:  1) If a function is monotone on an interval $[a,b]$, then $f(a)\le f(x) \le f(b)$ for $x\in[a,b]$. Therefore if there exists left-hand (right-hand) limit of this function at a given point, then it must be finite. Now we must show that a both left-hand and right=hand limits exist. We use the fact that if we take a sequence $(x_n)\rightarrow c^{+} \in [a,b]$ then $f(x_n)$ is bounded and monotone and therefore it is convergent. Why are springs shaped the way they are? Why are springs coiled the way they are? Why not some other shape? Is the shape due to its elasticity or something?