My question is: What is the result of this limit:   $\displaystyle \lim_{n \to +\infty} \frac{{n \choose n/2}}{2^n}=$ ?
For each positive even integer $n$, set   $$P_n = \displaystyle {{n} \choose {\frac{n}{2}}}\frac{1}{2^n}.$$  Show that $\displaystyle \lim_{n \to \infty} P_n$ exists and determine its value.   Here's what I have so far: each $P_n$ can be thought of as the probability of tossing a fair coin $n$ times, and obtaining exactly $\frac{n}{2}$ heads. My expectation is that $P_n \to 1$, since in any large trial, one would expect to record just about as many heads as tails. But I'm not sure how to mathematically justify this hunch.  Hints or solutions are greatly appreciated.
Prove $\lim_{n\to\infty} \frac{a_n}{n}$ exists for positive sequence where $a_{n+m} \leq a_n + a_m$ Let $\{a_n\}_{n=1}^\infty$ be a positive sequence of real numbers such that $a_{n+m} \leq a_n + a_m.$  Prove that $\lim_{n\to\infty} \frac{a_n}{n}$ exists by showing that $\lim_{n\to\infty} \frac{a_n}{n} = \inf_{n \geq 1} \frac{a_n}{n}.$   A similar question has been posted previously, see  , and I have found that this is the general result of "Fekete's Subadditive Lemma" for sequences, but I am unhappy with the few proofs I've seen.  Perhaps the clearest one I have found is  , but this seems a bit tricky for what is expected on this qualifier question.  We are given a hint "treat $\liminf$ and $\limsup$ separately".  Does anyone know of a more direct way to show that this limit exists? Is there a faster way to open a process as Administrator (With UAC enabled)? Currently (In Windows 10) - if I want to open a CMD prompt, I'll hit the Windows key, type CMD and hit Enter. If I want it open as an administrator, I have to right click the item and click Run As administrator. Is there a way I can do this without using the mouse? Why do the major and minor groove exist in DNA? So I know that the major/minor groove arise from the offset pairing of the two strands in the helix.  All my textbook tells me is that "pitch causes the off-set pairing of DNA and this gives rise to the major and minor grooves."  I'm not too sure what this is saying.  What exactly is pitch and how does it give rise to the grooves? Making Map Book of pages with same area shown but different layers turned on/off In ArcMap a Map Book is best suited for a series of features that each get their own page. It allows you to pick a feature and generate a Map Book Page for each feature including any additional information you wish to add to each page via dynamic text, charts, etc. Is there an alternative for say a series of pages showing the EXACT same area but with different layers turned on and off? We have a Capital Improvements document that I have to create showing the City Boundaries on every page. One page has all police and fire stations listed, another shows proposed water systems, another shows expressways and highways and work being done near or on them... etc., ad nauseum. Is there some way to set this up so that I DON'T have to created a separate MXD for each Map Page of this document? C# 3.0 Auto-Properties - Is it possible to add custom behaviour? I would like to know if there is any way to add custom behaviour to the auto property get/set methods.  An obvious case I can think of is wanting every set property method to call on any PropertyChanged event handlers as part of a System.ComponentModel.INotifyPropertyChanged implementation.  This would allow a class to have numerous properties that can be observed, where each property is defined using auto property syntax.  Basically I'm wondering if there is anything similar to either a get/set template or post get/set hook with class scope.  (I know the same end functionality can easily be achieved in slightly more verbose ways - I just hate duplication of a pattern) How does ownership in fractional shares work? I've been trying to find out some more information about fractional shares, and how they work. I understand conceptually, you own x% of a share rather than x "full" shares, but I'm curious about how such a system is implemented.  As I understand it, in a "traditional" purchase of shares, the role of a broker is to act as an executor on your behalf, and does not hold any assets themselves. In the case of a fractional share, is this then not the case? For example, the broker holds the complete shares, and then just ensures that they keep an internal record of each partial owner in their system? So there would be no external record of partial ownership?  Thanks! Tablet or Phone - Android What's the difference between  and ,  and ? If you're interested in grammar, as I am, I am sure you have delved into a thought process about infinitive to, and like me, you have probably questioned what it is, or what it could be defined as. My mind always wants to place it in the Eight Parts of Speech, and I know this is foolish of me to some regard, so I stray from it, but I still wish to designate it under something that can be explain it. So, this is the motivation for this post, this question: What is the infinitive to, and why?  Here are my thoughts: There are three real possibilities, going back through the times. First, we can say that it is part of the verb that comes after it (to go, to run, etc.). This could be supported by the claim that you are not supposed to "split" infinitives ("To boldly go"), but only that.   Secondly, we could consider it a subordinator. I came across this designation most recently in my studies of modern and transformational grammar. As of right now, it is still the one I accept, especially as a result of the trend of designating more phrases as clauses. It works, except it's weird. Do we call this a unique subordinator as a result of how it operates? For example, to-infinitive clauses call for non-finite verb forms, which typically have an implied subject as a part of their clause. For example, "I want to go to the store." In the infinitive clause, "I" is the implied subject, as a understand. So, again, this works. However, things get a little weird when other subordinators come into play. For example, "I'm excited for you to win the competition." Now, if I'm right, those of you will say that "for" is a preposition here, as a result of the new-grammar stuff, but I'm going to count it as a subordinator, as defined by my own beliefs and that of a lot of other grammars. So, how do the clauses work in such a sentence? According to transformational grammar, "for you to win the competition" is defined as an ordinary clause. So, does that mean that it also contains a secondary clause within itself (the infinitive clause part)? If so, how do those two clauses work in conjunction to each other when one is so integrated? "For you to win the competition" looks to just be one clause, but if that's the case, doesn't "to" have to represent something else?  This moves us on to the third way to look at things, which is to consider "to" as auxiliary verb, a defective one at that. There are a lot of things to support this, and it even allows for the split infinitive. Furthermore, it calls for a non-finite verb form to follow, like an auxiliary. But, as I said before, it's defective in the way that is all it can function as.  As I said, I still accept to as a subordinator, but I want to be proven wrong or right. Wrong by showing evidence of it being something else, or right by showing how clauses operate when it seems to be layered up. Is clockwise or counter-clockwise rotated text easier to read? When rotating text with css, should it be rotated clockwise or counter-clockwise?  Is one more readable than the other?  Counter Clockwise:   &nbsp;&nbsp;&nbsp;&nbsp;&mdash;&nbsp;&nbsp;&nbsp; Clockwise: How to increase difficulty in Dungeon World I've been GMing for a couple of sessions now and I'm still struggling at making encounters or situations more difficult. I will describe dangerous situations and encounters in fiction but the mechanics make it too easy for my players to succeed.  For instance no matter how big I describe the Troll, if the fighter of the party charges him and swing his sword, roll+2 for Hack and slash, he'll easily hit the troll. Almost 40% chance to hit without consequences. How do I get the number of days between two dates in JavaScript? For example, given two dates in input boxes:  &lt;input id="first" value="1/1/2000"/&gt; &lt;input id="second" value="1/1/2001"/&gt;  &lt;script&gt;   alert(datediff("day", first, second)); // what goes here? &lt;/script&gt; Change pivot or local origin of an object How can I change the local origin of an object, without changing the object location in the world space.  This is the object located at position (0,0,0) in the world space    This is the same object located at position (0,1,0) in the world space, but with its local origin offset so that the object is in the same place    Is there a way to do this in a single step? What's the meaning of the term "gib"? Especially while talking about FPS, the term "gib" is quite common. While I get it's related to a sudden and gruesome death, I'd like to know its actual meaning and where it comes from. What's your preferred way of getting current system status (current CPU, RAM, free disk space, etc.) in Python? Bonus points for *nix and Windows platforms.  There seems to be a few possible ways of extracting that from my search:   Using a library such as  (that currently seems not actively developed and not supported on multiple platform) or something like  (again no activity since 2007 it seems and no support for Windows). Using platform specific code such as using a os.popen("ps") or similar for the *nix systems and MEMORYSTATUS in ctypes.windll.kernel32 (see ) for the Windows platform. One could put a Python class together with all those code snippets.   It's not that those methods are bad but is there already a well-supported, multi-platform way of doing the same thing? I am sitting inside a closed room with a cement ceiling just above my head. The pressure of the atmosphere is exerted on the ceiling from above(outside). But solid cement not being a fluid does not transport that pressure into my room. But i still feel the atm pressure which has no way to be transported into the room.I do not boil to death or anything. How does this happen? What does Use an Object cover? I observe the following strange characters ^@ in syslog at the time when my machine freezes.  Then i have to hardboot my machine.  What could be wrong any ideas?  Oct 12 12:20:52 nanak-P570WM anacron[1074]: Normal exit (1 job run) ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@  Oct 12 12:24:28 nanak-P570WM kernel: imklog 5.8.11, log source = /proc/kmsg started. Oct 12 12:24:28 nanak-P570WM rsyslogd: [origin software="rsyslogd" swVersion="5.8.11" x-pid="543" x-info="http://www.rsyslog.com"] start HashSet vs. List performance It's clear that a search performance of the generic HashSet&lt;T&gt; class is higher than of the generic List&lt;T&gt; class. Just compare the hash-based key with the linear approach in the List&lt;T&gt; class.  However calculating a hash key may itself take some CPU cycles, so for a small amount of items the linear search can be a real alternative to the HashSet&lt;T&gt;.  My question: where is the break-even?  To simplify the scenario (and to be fair) let's assume that the List&lt;T&gt; class uses the element's Equals() method to identify an item. I am studying English in Korea.   All my friends and I are studying about 'sentence patterns'  Sentence patterns, said in here, are S+V, S+V+C, S+V+O, S+V+I.O+D.O, S+V+O+O.C.  But my teacher said these 5-sentence patterns are not perfect in grammar. But  because we don't use English by mother language, they will be helpful for understanding English.  Then, I came up with something.   Is sentence patterns used everywhere? (like U.S.A, U.K, Australia, anyother country where people use english by their mother language?)  I really wonder.