Show that $\log({n\choose n/2})=O(n)$, I've tried to use stirling's formula but didn't get any close
Evaluate   $$\lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n}.$$   $\underline{\bf{My\;\;Try}}::$ Let $\displaystyle y = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n} = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \left(\frac{(n+1)\cdot (n+2)\cdot (n+3)\cdots (n+n)}{(1)\cdot (2)\cdot (3)\cdots (n)}\right)$  $\displaystyle y = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \left\{\ln \left(\frac{n+1}{1}\right)+\ln \left(\frac{n+2}{2}\right)+\ln \left(\frac{n+3}{3}\right)+\cdots+\ln \left(\frac{n+n}{n}\right)\right\}$  $\displaystyle y = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \sum_{r=1}^{n}\ln \left(\frac{n+r}{r}\right) = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \sum_{r=1}^{n}\ln \left(\frac{1+\frac{r}{n}}{\frac{r}{n}}\right)$  Now Using Reinman Sum  $\displaystyle y = \frac{1}{2}\int_{0}^{1}\ln \left(\frac{x+1}{x}\right)dx = \frac{1}{2}\int_{0}^{1}\ln (x+1)dx-\frac{1}{2}\int_{0}^{1}\ln (x)dx =  \ln (2)$  My Question is , Is there is any method other then that like Striling Approximation OR Stolz–Cesàro theorem OR Ratio Test  If yes then please explain here  Thanks
How to get object size in memory? I need to know how much bytes my object consumes in memory (in C#). for example how much my Hashtable, or SortedList, or List&lt;String&gt;. How can I check if /bin/x86_64/bin/ls is a directory in a bash shell script  Here is what I tried:  #!/bin/bash  if [ $# -eq 2 ]; then     if [[ "$1" = /* ]]     then     cd ./bin/x86_64/bin/ls         if [ -d "$1" ]; then             echo "ok"             i="$1"             echo $i         else             echo "error2"             exit         fi     else         echo "error"         exit     fi fi Which is the simplest way to update a Label from another Thread?   I have a Form running on thread1, and from that I'm starting another thread (thread2).  While thread2 is processing some files I would like to update a Label on the Form with the current status of thread2's work.   How could I do that? Since I am a regular Stack Overflow follower, I came across this and was shocked a bit regarding the design and user experience perspective.  Attaching picture 1 - it states that I can click any site and view a synopsis about it. Clear!!!    Attaching Picture 2 - you can see that when I clicked Webmasters, the design wants me to scroll horizontally to view all information and basically it hides information.    It would be nice, if it did something like below : Why didn't Aragorn keep going with the army of the dead? I execute quite a lot of sudo commands.  I noticed that if I execute one and then another one within few minutes, the 2nd time I don't get the message:     [sudo] password for my_username:   but I do get it when there is more time between the execution of the two commands.  What's the period of time in which the message is not appearing? How can we check it / update it? Battle of Hoth: how did the AT-ATs manage to disable the shield generator? (This question is related to , and was originally posted as an answer there, but it's not the same)    In , the Empire attacks Rebel Echo Base located in Hoth. The Rebels are alerted before the attack, and raise some sort of shield to prevent orbital bombardment of their base. The Empire is then forced to deploy land forces to first disable the shield generator, before destroying the base. Presumably, the Imperial forces are deployed using dropships.  My question is about how the Rebel shield generator works, and how the Imperials manage to bypass it.  I think the shield generator is something of a plot hole. At first I incorrectly thought it was a planetary shield (i.e. a shield around all of Hoth) preventing orbital bombardment, and also preventing any dropships (with their AT-AT payload) from landing, but I think : the  states it is an energy field around just an area of Hoth, presumably Echo Base:     VEERS: My lord, the fleet has moved out of light-speed. Com-Scan has   detected an energy field protecting an area around the sixth planet of   the Hoth system. The field is strong enough to deflect any   bombardment.   So my initial objection ("how did the Imperial dropships bypass the shield?") has been answered: they didn't. They landed outside the energy field to deliver the AT-ATs. But now I have another question: does an AT-AT have to walk inside the field to fire at the generator, or can it fire into the field from outside?  Maybe the field simply cannot stop matter, only energy weapons (therefore preventing orbital bombardment), in which case the AT-ATs would be able to just walk inside. However, the script also states:     LEIA: All troop carriers will assemble at the north entrance. The   heavy transport ships will leave as soon as they're loaded. Only two   fighter escorts per ship. The energy shield can only be opened for a   short time, so you'll have to stay very close to your transports.   So it's clear that the energy field is also capable of stopping matter, in this case troop transports and fighters. It follows that it should also prevent AT-ATs from walking through it. So the only way I can see the walkers disabling the generator is by firing into the energy field!  This would mean the field is strong enough to stop matter such as ships and fighters, and energy weapons used for orbital bombardment (and also the Ion Cannon the Rebels use as an anti-orbit battery: Leia also states they must lower the shield to use it), but not able to stop whatever energy weapons the AT-ATs use!  So how on Earth did Imperial Walkers manage to disable the shield generator? When I gain control of a creature, do I also gain control of any auras or equipment attached to it? What would happen if I 'd a creature that had been enchanted by a bestow creature and cast a spell to give it protection from the enchantment's color on it to remove the bestowed creature?  For instance:  They have just bestowed a  on a  and passes the turn. I use Portent of Betrayal to take control of the Asphodel Wanderer, attack and in the second main I cast Gods Willing to give the Asphodel Wanderer protection from black. Nighthowler detaches and where does it go, my control, opponents control or graveyard?  As an additional question, in the same vein:  What if I destroyed the creature while it was on my side? So change the gods willing to a hero's downfall in the example.  Relevent (under state-based actions):     704.5n If an Aura is attached to an illegal object or player, or is not attached to an object or player, that Aura is put into its owner's graveyard. How many ways can $r$ nonconsecutive integers be chosen from the first $n$ integers? I need to (permanent) redirect all of the following:   www.example.com www.example.com/folder sub.example.com sub.example.com/folder   Basically, all URLs that are reachable without being intended to: www.new-example.com, I want to do this within Apache and using the mod_rewrite module via the .htaccess file. Deciphering the RSA encrypted message from three different public keys I have three different 1024-bit public keys with common exponent $e$ but different moduli. A message $m$ is encrypted (without padding) using the three keys, which results in three different encrypted messages.  Given the three pairs of public keys $(N_i,e)$ and the encrypted messages $c_i=m^e\bmod N_i$ , how do I decipher the original message $m$ ? Cooking away alcohol When you cook with wine or spirits, when does the alcohol cook away? Obviously high temperatures will do it, but how low of temperatures will work? Also, does it vary by the type of alcohol? KeyListener not working Proving a triangle with different edge colors exists in a graph. This is again some homework translated (hopefully not too badly) from my book  The graph $K_{n}$ is colored using $n$ different colors, in a way that each color is used at least once. Prove that there exists a triangle with its edges colored in different colors. How to call shell commands from Ruby Ruby Style: How to check whether a nested hash element exists Convince grep to output all lines, not just those with matches Say I have the following file:  $ cat test  test line 1 test line 2 line without the search word another line without it test line 3 with two test words test line 4   By default, grep returns each line that contains the search term:  $ grep test test  test line 1 test line 2 test line 3 with two test words test line 4   Passing the --color parameter to grep will make it highlight the portion of the line that matches the search expression, but it still only returns lines that contain the expression. Is there a way to get grep to output every line in the source file, but highlight the matches?  My current terrible hack to accomplish this (at least on files that don't have 10000+ consecutive lines with no matches) is:  $ grep -B 9999 -A 9999 test test     If grep can't accomplish this, is there another command-line tool that offers the same functionality? I've fiddled with , but it doesn't seem to have an option for it either. How do I use SQL's GETDATE() and DATEADD() in a Linq to SQL expression? If I have a Linq to SQL expression like this:    from subscription in dbContext.Subscriptions  where subscription.Expires &gt; DateTime.Now select subscription   I want this to to use the SQL Servers GETDATE() function instead of the time of the machine running the C# program.  The next question would be how to translate this:  DateTime.Now.AddDays(2)   to this:  DATEADD(dd, 2, GETDATE()) Is there a term denoting the writing of words using numbers instead of letters? I am beginning with Abstract Algebra and I'm trying to understand Lagrange's Theorem. The theorem reads      For any finite group $G$, the order of every subgroup $H$ of $G$ should divide the order of $G$.   It seems simple and I've used it to solve some exercices but I believe I'm missing the essence of it. Is there an example that can help me understand it better, or visualize it? Is there a geometric interpretation?