$N$ is a positive integer. I want to calculate this limit, but i couldn't get anywhere when i tried.  $$P[n]=\left(1+\frac {1}{n^2}\right)\left(1+\frac {2}{n^2}\right)\cdots\left(1+\frac {n-1}{n^2}\right)$$ as $n\to \infty$  I tried to apply $\ln()$ at both sides to transform into a sum etc.. tried to use functions to limit the superior and inferior intervals of the function like this: $$\exp\left(\left(n-1\right)\ln\left(1+\frac1{n^2}\right)\right)&lt;P[n]&lt;\exp\left((n-1)\ln\left(1+\frac{n-1}{n^2}\right)\right)$$ The only thing I get is that: $1&lt;\lim(P[n])&lt;e$  Can anyone help me to solve this?
How can we compute the following limit:     $$\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$$   Mathematica gives the answer $\sqrt{e}$. However, I do not know to do it.
-ing vs -in' ending I wonder if the &quot;g&quot; in the -ing forms is pronounced. When I hear it it seems as if it's not pronounced sometimes or just slightly, though sometimes I've been told that I should pronounce &quot;g&quot; for example in &quot;meeting&quot; just to avoid saying &quot;mitten&quot;. So how should I pronounce &quot;-ing&quot;? Sometimes -ing is written in an informal way as -in' such as:  taking takin'  Is the letter &quot;g&quot; in each case pronounced differently? Proving that $(3n)!$ is divisible by $n! \times (n + 1)! \times (n + 2)!$ if $n$ is greater than 2 Does $\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(...\cos x))}}$ exist? Possible Duplicate:         Does the following limit exist?  $$\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(...\cos x))}}$$  If yes, find the limit.  If no, please explain why the limit doesn't exist.  I think, the limit exists.  So, I tried to use Squeeze theorem but didn't work. In Avengers: Endgame,       Thanos was able to stop and catch Mjolnir and Stormbreaker mid-air as both Captain America and Thor used/called them BEFORE he got the Infinity Stones.   Since this character is clearly not worthy and had no super powers at the time, how were they able to wield Thor's Mjolnir and Stormbreaker, when even the Hulk couldn't do it in The Avengers? What is a "slug" in Django? How to stop Gedit, Gvim, Vim, Nano from adding End-of-File newline char? Is there a Google Drive client available? Invoking makeindex for nomencl I have spent many many hours failing to implement nomencl for my PhD thesis. The problem seems to be with invoking makeindex and creating the .nls file  I keep reading that I must invoke makeindex or write a Unix command:  makeindex filename.nlo  -s nomencl.ist -o filename.nls   But I don't know how to do this. I would be grateful for some instructions i.e. where do I type the command?    I am using TeXShop 2.43. How to get layer extent in OpenLayers or zoom to WMS layer extent? I have added WMS layers using follow:  basemap = new OpenLayers.Layer.WMS( "Layer Name1", "http://localhost:8080/geoserver/wms/workspacename", {layers: 'workspacename:layername'} );   utility = new OpenLayers.Layer.WMS( "Layer Name2", "http://localhost:8080/geoserver/wms/workspacename", {layers: 'workspacename:layername', transparent:"true"}, {isBaseLayer:false} );   Now I want to get zoom to my utility layer, How can I?  I have tried as following:  map.zoomToExtent(utility.getDataExtent()); Birthday problem: why is this solution wrong? This question is about the birthday problem: the probability that in a group of n people, at least two of them have the same birthday ().  An easy way to calculate the probability is to calculate first the probability that no two people have the same birthday.  Let's say that I want to calculate the probability that in a group of 20 people, NO two people have the same birthday.  So, for 20 people with different birthdays, I can choose the first birthday in 365 ways, the second in 364 ways and so on, while for 20 people who can have the same birthday, I can choose the first birthday in 365 ways, the second also in 365 ways, and so on. At the end:  $p={365 \cdot 364 \cdot 363 \cdot ... \cdot 346 \over 365 \cdot 365 \cdot 365 \cdot ... \cdot 365}\approx 0.59$  This is the right method and I understand it. I don't understand why the following method is wrong:  The probability that in a group of 20 people NO two people have the same birthday is the ratio between the combination without repetition of 20 birthdays and the combination with repetition of 20 birthdays:  $$p={C_{365,20} \over C'_{365,20}}={\binom{365}{20} \over \left(\binom{365}{20}\right)}={\binom{365}{20} \over \binom{365+20-1}{20}}=\frac{{{365!}\over {20!\cdot(365-20)!}}}{{(365+20-1)!}\over{20!\cdot{(365-1)!}}}\approx 0.35$$  I understand the mistake is in the denominator (as the numerator is the same of the other method, after simplifying that 20!), but why? Isn't it right to calculate the k-combination with repetition of 20 birthdays?  Thanks for helping! When is 6 a square in Zp To find when $6$ is a square I divided $6$ in to $2$ and $3$ by The Legendre symbol.  $$\frac{6}{p} = \frac{2}{p} \times \frac{3}{p}$$  I know that we can multiply the cases of when $2$ and $$ are squares because then it is (1)(1) by Legendre.   $2$ is a square when $p \equiv 1 \pmod{8}$   $3$ is a square when $p \equiv 1 \pmod{12}$  so $6$ is a square when $p \equiv 1 \pmod {24}$ ($96$, but the $lcm(8,12)= 24$)   but how do I find the cases when $2$ and $3$ are both NOT squares? because then $6$ would be a square then as well, $(-1)(-1) = 1$ by Legendre. Is it possible to dynamically set Views' pager settings? Say for instance, you have a view that displays when viewing a certain node type, but you want the pager settings to be different for each individual node. I was thinking of making a CCK field like "pager setting" and then give it an integer option of x-y. But I don't know if it's possible to somehow dynamically plug that field into the Views' settings. Or is there another way to do this maybe? How to to calculate the hash of an unordered set Suppose I have a set of elements, with known hash (e.g. SHA-2). How can I calculate the hash of the set? With it I mean an unordered set, so the order of elements is undefined and shall not play any role in determining the hash of the set.  In theory any commutative operation applied to the hashes should be fine, but I have the strong feeling that XORing all of them together is quite unsafe. An attacker could add an element to the set to produce a hash collision, although finding an element with that wanted hash is as hard as breaking the hash of the whole set.  Anything better solution than the trivial one of sorting the elements into a list, calculating their hashes and then calculating the hash of the hash list?  I think Merkle trees can't help here if they are based on some sort of ordering of the elements. How to draw good arrows explaining parts of equation? Keyboard macro to change default audio device Windows 7, Logitech G110 keyboard  2 audio devices (headphones and speakers)  I am hoping to find a quick way to macro a key to switch to headphones, and a key to switch to speakers.  Anyone have a good program or anything of that nature?  EDIT:  Using AHK to try and get this task working as hoped is sadly not doing the trick.  I have included a screenshot of my audio set, as well as a screenshot of my script as it sits now.    I have also tried to uncheck "show disabled devices" and modified the script to be Down 1, and Up 1 respectively.  It will work for one device, but it refuses to switch to the secondary. Why is the B field of a solenoid equal to $\mu_0 i n$ while that of a loop is $\frac{\mu_0 i R^2}{2(R^2+z^2)^{3/2}}$? How will learning about philosophy impact real-life? Examining the practical, impactful, real-life benefits of getting a degree in philosophy It is often asked, &quot;What is the practical use of philosophy?&quot; or &quot;How will learning about philosophy impact real-life?&quot; It is easy to see why there is such mystery surrounding the topic. Most schools up through high school may only offer 1 or 2 courses on it, if any. It is also often talked about in general conversation as a &quot;wishy-washy&quot; subject, often negatively, as though it contains no answers but only raises an endless series of questions. To add to this, those who hear of the subject but know very little of it may be surprised to see it offered at their university as a major. It is clear to most why some people would want to major in Computer science, Mathematics, or Physics. But what would a degree in philosophy get you? X,Y,Z missing in F3 When I press F3 to see the debug screen in I use Minecraft 1.8.1, some information is missing.   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