Compute limit $\lim_{n\rightarrow\infty}\frac{1*3*5*...*(2n-1) }{ 2*4*6*...*(2n)}=0$
To show for following sequence $\lim_{n \to \infty} a_n = 0$ where $a_n$ = $1.3.5 ... (2n-1)\over 2.4.6...(2n)$
I am trying to solve an integral in $k$-space where I want to calculate the ratio of the average energy of an arbitrary electronic band ($E(k)=k^n)$ to the square of average group velocity of electrons($v_g   = k^{n-1})$ approximately in the form of this expression given by $$L=\frac{\frac{\sum_{\mathbf{k}}k^{n}}{\sum_{\mathbf{k}}1}}{\left(\frac{\sum_{\mathbf{k}}k^{n-1}}{\sum_{\mathbf{k}}1}\right)^2}\;.$$  Now, I want to do this for $d$-dimensions, so I convert this sum to integral.  $$L=\frac{\dfrac{\int_{0}^{k_f}k^{n} d^d k}{\int_{0}^{k_f}1d^d k}}{\left(\dfrac{\int_{0}^{k_f}k^{n-1}d^d k}{\int_{0}^{k_f}1d^d k}\right)^2}$$ where $d^dk  = \dfrac{d \pi^{d/2}}{\Gamma(\frac{d}{2}+1)}k^{d-1}dk$ is the volume element in $d$-dimensional space.  I want to get a simple expression for $L$ and how it depends with $k_f,n$ and $d$. any physical insight into this integral could also be helpful. "I'm right, aren't I?" My dictionary reports that aren't is also an abbreviation for am not, when used in questions.  Is this use of aren't common? If it is common, are there any differences between the different English dialects?     Why aren't I being given a pay raise? How do I disable DVD auto eject in Windows 7? I just discovered an incredibly annoying "feature" of Windows 7 tonight.  If you click on your optical drive in Explorer, and there is no disc in the drive, Windows 7 is kind enough to open the tray for you.  I'm finding it hard to contain my rage over this idiotic design decision.  WTF?  According to , this behavior was apparently introduced in Vista.  The author posted a workaround, but I don't want a workaround.  I want it to stop doing this.  Like he, I have a case with a door that covers the drive bays.  If it's closed, and I accidentally click on that optical drive, it knocks on that door 10 or 12 times before giving up.  Can this behavior be turned off in Windows 7? Is the word "data" now considered singular? If so, what about "datum"? I'm in trouble with time dilation: Suppose that there's two people on the Earth (A,B), they are twins and each other has a clock. (So they are at the same reference frame). B travels in a spaceship and is orbiting around the Earth, as B's speed has increased there's some time dilation. So it's supposed that if A "looks" to B, B's clock will run slower than A's, and viceversa. And if one day B decides come back to the Earth, he will be younger than his twin and the hour of his clock would be different (earlier than A's clock) because of time dilation.  My first question is, why each person see the other's clock as the slow one? (In fact, I found some information that may solved it but I'm not sure if it's the right answer, and if it is, I don't understand it. -> It's because everyone thinks about himself as the reference frame. In other words, that for B, he is the stationary and who moves is A, so he sees A's clock run slower as B is "stopped" and the same for A. B is who is moving and A the stationary.)  The other question I have is that if it's true that each one sees the other as the slow one there's something that i missed out. We all agree that B (after going to the space and came back to the Earth) he is younger than his brother. So, why B don't see A moving and aging very fast if A see his brother and his things going very slow? Let's imagine they were looking to each other all the time, if both of them see the other going slowly once they meet and "discover" B is the young one... how is that possible?  [I think that in one episode of Cosmos of Carl Sagan (though may be I'm mistaken) he said that a neutrino "borned" in the Big Bang could have seen the creation of the Universe until today in few seconds due to its high speed, here starts my doubts: or i misunderstood something or there's contradictory information]  (I don't know anything about physics, only what is taught at high school as I'm a teenager so it'd be better to answer with no kind of calculation as I'd not understand it.) Is there a comma needed before quotation? Sam developed an annual sweepstakes promotion entitled “The Free Gas Punt Return Game”.    Do I need a comma after entitled or not? I'm new to Ubuntu. What's the application extension for development reasons? I know Windows is mainly .exe and Mac is .dmg or .app. Does Linux have an unique one? How can I get the size of an Amazon S3 bucket? I am trying to build a computer, placing an  in a custom .  The case is an extra small form factor which can accommodate the Mini ITX board, but it requires an external power supply, similar to the power-bricks that laptops use.  I am looking at different power supplies and like , but I cannot figure out whether/how to power this system, because they provide a 24-pin connector, but the motherboard has a 4-pin connector as well.  Is the 4-pin connector optional or is it required? If it is required, how can I provide power for this system? If $f: \mathbb Q\to \mathbb Q$ is a homomorphism, prove that $f(x)=0$ for all $x\in\mathbb Q$ or $f(x)=x$ for all $x$ in $\mathbb Q$.  I'm wondering if you can help me with this one? "Z-level" in TikZ iOS Calendar Access Permission Dialog, force it to appear? can u suggest me any short trick of solving these kind of problems. How to find last four digits of any number raised to some power ? $n$ choose $k$ where $n$ is less than $k$ I am working on parameter estimation and one of the estimators involves a summation of $_nC_k$ ($n$ choose $k$) expressions. For some iterations, I need to compute expressions like $_0C_1$, $_0C_2$, etc.  In general how do we compute $_nC_k$ when $n$ is less than $k$? Do we still use the formula $\frac{n!}{(n-k)!k!}$ and use the gamma function to compute the negative factorial?  Thanks! Can Metamagic be used in Item Creation? What are the limits? One of my players has recently obtained the Craft Wondrous Item feat. In an attempt to prepare for what he may build I want to know the specific limits of Wondrous Items.  I know that someone can create an item for which they do not meet the prerequisites by adding +5 to the spellcraft DC, and that assistance is a thing.  Can someone use meta-magic when crafting a non-specific wondrous item?  Example: Amulet of Empowered Fire-Ball.  If so, can they use meta-magic feats they do not know, substituting for +5 DC?  Can the Spell Level exceed 9 for these items, as long as they add +5 for not meeting the CL requirements, and make the higher check?  Example: Amulet of Maximized Empowered Still Silent Disintigrate. What's wrong with this demonstration?:  $$A \iff 1 = 1^1$$ $$A \implies 1 = 1^\frac{2}{2}$$ $$A \implies 1 = (1^2)^\frac{1}{2}$$ $$A \implies 1 = ((-1)^2)^\frac{1}{2}$$ $$A \implies 1 = (-1)^\frac{2}{2}$$ $$A \implies 1 = (-1)^1 = -1$$  So finaly we get  $$1 = -1$$ There are $k$ penguins, $k\ge 3$. They are all different heights. How many ways are there to order the penguins in a line, left to right, so that we cannot find any three that are arranged tallest to shortest (in left to right order)? The penguin triples do not have have to be adjacent.  This problem is courtesy of someone I know only as "DaMancha." Mount USB drive with write permissions for everyone or specific user Accessing static member through invalid pointer: guaranteed to "work"? Is there a way to make sure that a deleted file can not be recovered?   As you know, deleting a file by pressing shift-del or using trash doesn't mean that file is gone forever. It resides somewhere in the computer.  In day to day life, law enforcement or thieves use recovery programs to reveal our private data or something else.   They can recover all data that resides on hard disk, in RAM, and even USB.  To protect us from these people, what should one do? Which program should one use?  Note: I don't want an answer like first use a recovery program then erase that file with shred. With just one program, I want to erase all unused/deleted files on my computer.