Formal way to proof that $k! > (k/2)^{(k/2)}$ for any $k \in \{1,2,3,4...\}$
How to show that for all k, $k! \ge (k/2)^{k/2}$
What are the practical applications of the Taylor Series? I started learning about the  in my calculus class, and although I understand the material well enough, I'm not really sure what actual applications there are for the series.  Question: What are the practical applications of the Taylor Series? Whether it's in a mathematical context, or in real world examples. Enabling GPU rendering for Cycles? How to monitor process/program execution in windows? We are trying to develop a small application that can monitor the programs/processes that are executing in a windows machine.  If the program/process is not supposed to run, it should be blocked. It works similar to an antivirus.  This is the basic idea.  I want to know the ways to hook into the OS to get notified about every single program/process trying to run in the machine. Is there evidence of "think tank" meaning "the brain"? The  Random House Dictionary defines the expression  as:         a research institute or organization employed to solve complex problems or predict or plan future developments, as in military, political, or social areas.      According to  the above connotation  is an extention of an earliar sense  which referred to "the brain":        also think-tank, 1959 as "research institute" (first reference is to Center for Behavioral Sciences, Palo Alto, Calif.); it had been colloquial for "the brain" since 1905.      : shows usage of "think tank" only from the late '50s.  Is there evidence  of earlier usages of "think tank" meaning "the brain"? Despite  its colloquial   usage I assume  the expression was quite widespread before it was adopted by the Stanford University. How to escape a single quote? This works fine:  sed -i 's#    @driver.find_element(:xpath, "//a\[contains(@href,##' temp_spec.rb   against a source of  @driver.find_element(:xpath, "//a[contains(@href,'change_district')]").click   I am just left with:  'change_district')]").click`   but when I try to add a single quote at the end it fails:  sed -i 's#    @driver.find_element(:xpath, "//a\[contains(@href,\'##' temp_spec.rb  syntax error near unexpected token `('   I am using \' to escape the single quote.  Note that I am using a # as a delimiter instead of the normal / delimiter as there are /'s in the text. Why is spectrum obtained by sunlight, said to be continuous? My teacher spoke about atomic spectra today, and he explained that, unlike the spectrum obtained by analyzing the sunlight, the spectra of atoms are not continuous.     I have a question about this - even the sunlight has the radiations emitted by the atoms of the elements which compose the sun, still spectra is continuous, which is in opposition to the statement that atomic spectra is discontinuous. So, spectrum obtained by sunlight is continuous even though it atomic spectra. In order to account for spectrum obtained by the sunlight to be continuous and atomic spectra to be discontinuous, can we confirm that sun consists of all those elements (sodium, helium, neon, mercury, etc) which emit the colors of frequency belonging to visible region? Finitely generated group which is not finitely presented Is there any easy group theoretical way of showing that the wreath product $G$ of two infinite cyclic groups is not finitely presented?  I was looking for a finitely presented group with a central subgroup isomorphic to the free group of countable rank and whose factor group is isomorphic to $G$; in this way assuming that $G$ is finitely presented we get a contradiction.   Any ideas? How to Fix "Read-only file system" error when I run something as sudo and try to make a folder/file? How is it possible that quantum phenomenons (e.g. superposition) are possible when all quantum particles are being constantly observed? I don't understand how quantum mechanics (and therefore also quantum computers) can work  given that while we work with quantum states, particles that this quantum state consist of cannot be observed, which is the most fundamental requirement.  If I am not mistaken, by "observed" we mean interaction with any other particle (photon, gluon, electron or whatever else). So my very important questions:   Aren't the particles this quantum state consists of interacting with each other? Why doesn't that cause the state to collapse? Aren't all particles in the universe interacting with Higgs field and gravitons etc? Why doesn't that cause every quantum state to collapse?   I feel there is something very fundamental in quantum mechanics that I am not aware of, hence I would be very pleased to have these questions answered. 3D bodies in TikZ Is there like a library that has default geometric figures like the ones shown below. Am not acquainted yet with 3D diagrams in TikZ. What is wrong with this substitution? I understand that the mechanics and design of a lightsaber aren't that complicated.  Do others besides the Jedi and Sith (or their minions) make or use lightsabers?  And if they don't, why not? I have Ubuntu 14.04 and two monitors. When I open applications I need some of them to be on the primary monitor and others On the secondary. At the moment, apps always open in the primary monitor.   I have dash icons only on one (left) display, so I can't launch apps using the cursor or active window setting from Compiz. Have Peter Capaldi's previous acting roles in the Doctor Who universe ever been addressed? Similar to this question posted in August 2014:       Has the fact that  (as  in The Fires of Pompeii, and  in Torchwood’s Children of Earth season) yet been addressed/explained either in, or out of universe? Where can I find the restrictions and easements that apply to a given UK Rail Ticket? UK rail ticketing isn't always the simplest thing in the world, but the basics aren't too hard to learn. There are  tickets valid at any time of the day, those available only outside of peak hours (typically after 9 and perhaps not in the evening rush hour), tickets only available at quiet times, tickets valid only on one train. These are mostly, but not always, valid in flavours of single (one way), day return (out and back the same day-ish), and (period) return (out on a given day, back within a month). That's it broadly, but you'll notice there's a bit of variation in even just those...  Next up, there are rules on what sort of route you can take. These largely revolve around , not doubling back, direct trains normally being fine, and not taking too long a route if there's a choice. (There's a whole art to making the most of that though, but this isn't the question for that!)  On top of all that, certain routes and/or stations will have additional restrictions, or easements where default restrictions are lifted. For example, one station might be in an area where the default rule is "off peak tickets valid after 9am", then has an additional restriction "off peak tickets to London only valid on trains arriving after 10am", and an easement "off peak tickets to stations along the insert line name here valid after 08:45".  By looking up these restrictions and easements, it's sometimes possible to save money on your journey. In the example above, you might assume you'd need a peak time ticket to travel on the 08:54 to a station along the local branch line, but the easement actually allows you to travel on a cheaper off-peak ticket instead.  How do you go about looking up what easements apply to a given ticket for a journey, and equally what extra restrictions (eg evening peak times or not-valid-via) apply? Big O, how do you calculate/approximate it? For all $x,y∈\Bbb{R}$ define that $ x\equiv y$ if$ x^2=y^2$ For all $x,y\in\Bbb{R}$ define that $x\equiv y$ if $x^2=y^2$ . Then $\equiv$ is an equivalence relation on $\Bbb{R}$ , there are infinitely many equivalence classes, one of them consists of one element and the rest consist of two elements.  Solution  True. To show that $\equiv$ is reflexive we need to show that $\forall x\in\mathbb{R} :x\equiv x$. Let $x\in\mathbb{R} , x\equiv x \mbox{ if } x^{2}=x^{2}$, which is obvious.  $[x] ={y\in\mathbb{R} |x\equiv y} =[0]={y\in\mathbb{R}|0\equiv y}={0}$. Hence $y^{2} =0^{2} =0$ which implies that $y=0$.  Can anyone please give me correct answer to this question. Can any one tell me the advantage of synchronized method over synchronized block with an example? I've got a Windows 8 (not .1, not Pro; OEM with a Lenovo Z580 laptop), which I want to use for WP8 development.  I've read  and , and a few more articles.  I have enabled in BIOS what looks like it has to do with hardware virtualisation - "Intel Virtual Technology" (is this VMX?). No signs of EPT or XD.  I've got the four Hyper-V items in the enumeration that msinfo32 shows.  However, in the 'Control Panel/Programs and Features/Turn on|off windows features' list, I don't have Hyper-V.  Is this available at all on my machine, and if not what can I do to get it working?  Thanks in advance. When is $f=\left(\frac{\sqrt{p+4}}{1-p}-1\right)x^5-3x+\ln5$ decreasing $\forall\; x$? When is $$f=\left(\frac{\sqrt{p+4}}{1-p}-1\right)x^5-3x+\ln5$$ decreasing $\forall\; x$?    Diffrentiating: $$f'=5\left(\frac{\sqrt{p+4}}{1-p}-1\right)x^4-3$$ If $f$ is decreasing, $f'&lt;0$: $$5\left(\frac{\sqrt{p+4}}{1-p}-1\right)x^4-3&lt;0\implies x^4&lt;\frac3{5\left(\frac{\sqrt{p+4}}{1-p}-1\right)}$$ Since if $\left(\frac{\sqrt{p+4}}{1-p}-1\right)\to0^+\implies x^4&lt;+\infty$, which holds forallx, So:$$p=\frac{3\pm\sqrt{21}}2$$which doesn't ply with textbook-answer, since it mentions an domain for p.