Show that $$\binom{n}{m}\le\left(\dfrac{en}{m}\right)^m$$ where $0&lt;m\le n,m,n\in N^{+}$  My idea: since $$(\dfrac{n}{m}-\dfrac{m-1}{m})(\dfrac{n}{m}-\dfrac{m-2}{m})\cdots\dfrac{n}{m}\le\left(\dfrac{en}{m}\right)^m$$ $$\sum_{i=1}^{m}\ln{\left(\dfrac{n}{m}-\dfrac{m-i}{m}\right)}\le m\ln{\dfrac{en}{m}}$$  then I can't,Thank you
I remember seeing an upper bound for the binomial $\binom{n}{k}$ with an exponential function, something like $\binom{n}{k}\leq \left(ne/k\right)^k$. What exactly is it, and are there other similar good upper bounds for $\binom{n}{k}$?  Edit: As the link in Macavity's comment shows, the bound is indeed $\binom{n}{k}\leq \left(ne/k\right)^k$. How can we prove this?
Headphone jack not working? How do I remap the Caps Lock and Ctrl keys? I'd like to remap my keys such that Ctrl behaves as the Caps Lock key and vice-versa. Also, if possible I'd like the settings to be available only within the currently logged-in user. How can I achieve this?  I'm a vim user, unlike the other people who use this machine, so I'd like those settings only for my account. US tourism/business visa from non-resident country I applied for multiple visas before (not US visa), but all of them were from my residence country. In all of these cases, I can get official paper which support my evidence (bank statement, letter which states I am employed with high salary, marriage certificate, security clearance, ...).  It is the same question as , but I am not residing; I have a business visa.  case-specific details: I am from Egypt, I have business visa for Ireland. While in Ireland, I want to apply for US business visa.  Why not to apply from Egypt? Because the waiting time is too long (75 days). The waiting time in Ireland in only 18 days. Both visits (Ireland, US) are short visits (3 weeks for each). Empty Disk when trying to install dual-boot system I recently purchased an SSD to speed up my computer experience. Before, I had Windows 7 and Ubuntu in a dual-boot system. The plan was to install Windows 8 on the SSD and then set aside ~30GB for the system files of Ubuntu.  I installed Windows 8 just fine on the SSD, then I booted the Ubuntu install CD to make my partitions like always - but GParted tells me the entire SSD is unallocated. Now I am afraid I will lose my Windows installation if I try to do anything to it.  Why does GParted think there is no partition on the SSD? Shouldn't it be 4 Partitions, like the Windows installer told me?  And is it possible to ignore this and install Ubuntu on the hard disk (where it, frankly, already is) and somehow make it possible to install GRUB on the SSD to revert my dual-boot system without damaging the installed Win8? prove that : $ i = \sqrt {-1}\ $ i have a pretty nasty question. i was glancing through a few olympiad papers and stumbled upon this question:  prove that  $ i = \sqrt {-1}\ $.  i tried the conventional methods namely euler's formula  but could not figure what to do next. how do you actually proceed further? i have checked a few books and a proof is not available.are there any proof available for this If $a\neq 1$, find $(a+1)(a^2+1)(a^4+1)\ldots(a^{2^n}+1)$. If $a\neq 1$, find $$(a+1)(a^2+1)(a^4+1)\ldots(a^{2^n}+1)$$   Or i.e.     If $a\neq 1$, find $\prod_{i=0}^n(a^{2^i}+1)$.   It really does seem like $$(a+1)(a^2+1)(a^4+1)\ldots(a^{2^n}+1)=a^{2^n+2^{n-1}+\ldots+2+1}+a^{2^n+2^{n-1}+\ldots+2+1-1}+a^{2^n+2^{n-1}+\ldots+2+1-2}+\ldots+a^2+a+1$$  Or i.e. $$\prod_{i=0}^n(a^{2^i}+1)=\sum_{i=0}^{2^n+2^{n-1}+\ldots+2+1}a^i$$  Is there an interesting proof of this? I'd appreciate any help. Does $\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x)\,dx\,$? If $\displaystyle\{f_n(x)\}$ is sequence of continuous function on $\mathbb{R}$ converging uniformly to $f(x)$, then does $\displaystyle\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) dx\,$ hold? I know  it holds if we integrate over an interval of the form $[a,b]$ but not sure about improper integral! What happened to these edits (and the associated reputation)? My reputation rating on SO took this hit an hour ago, and it is due to what appears to be the simultaneous deletion of multiple questions.  I am uncertain why it happened, am concerned, and would appreciate an explanation.  Going to  shows that it was removed due to moderation.  Was a script run deliberately, or were we hit with a question deletion attack; what happened?  -12     today -2  1 hour ago  removed     How to send a 2d Vector over UDP in c++ -2  1 hour ago  removed     How do I use the Eclipse debugger to debug a Java application? -2  1 hour ago  removed     PHP string error; inconsistency with local and server -2  1 hour ago  removed     Unable to start the MySQL service after a server migration -2  1 hour ago  removed     How to use Java to print a text file on a ticket printer? -2  1 hour ago  removed     How can all of the ZBrush Obj files be imported to make a complete model? In Shakespeare's Macbeth, the Third Witch says to Banquo: "Thou shalt get kings, though thou be none" (1.3.65); if I want to quote that line and write it in the third person singular, how should I conjugate the verbs "shall" and "be" correctly? Vueling is currently experiencing , with many flights delayed by several hours. Assuming I'm at the airport and my flight is certainly delayed by more than 3 hours, can I simply book another flight and claim compensation later?  According to EU's :     In addition, if you are denied boarding, your flight is cancelled or arrives more than 3 hours late on arrival at the final destination stated on your ticket, you may be entitled to compensation of EUR 250 - 600, depending on the distance of the flight.   But it's not clear if I actually have to board the flight or if simply having a purchased ticket is sufficient. I presume it would be possible to negotiate with Vueling representatives at the airport, but I'd rather not wait in the inevitable 100-person line to their counter if I'm covered by EU law. Can the determinant of a matrix can be made $0$? An entry of an $n \times n$ matrix with nonzero determinant is defined as interesting if by changing this entry (and only this entry) the determinant of the matrix can be made $0$.   Is it true that each entry of every matrix with nonzero determinant is interesting? Is it true that there is an interesting entry in each row of a matrix with nonzero determinant? Landscape topography from grayscale image Should a gas water heater be shut down if the water main has to be turned off? I have a brand new AO Smith 50gal gas water heater. Today there was a burst pipe from the cold spell we're having currently, so I had to turn off the water main to the house until the plumbers get here tomorrow.  With that, nobody had used any hot water all day so it should be full, but do I need to turn the water heater off until the water can be turned back on tomorrow to avoid any damage etc? When are attributive nouns plural? Sorry for the title, it is not very evident and intuitive but I really do not how to tell it better...  Well, you know, several times, or better, many times, we use this form:  If I want to say:      "development of special weapons was   the first point in Hitler's   program..."   I will say this (a better form):      "special weapon development was..."   OK...  is it      "special weapon development"   or      "special weapons development" (note   the plural...)   what's the grammar rule in order to understand how to use this very used form? How do constant aperture zoom lenses work? I am reading Velleman's "How to Prove it", and in one example he proves  $\neg \exists x P(x)  \implies \forall x \neg P(x)$    in the following way:    Suppose $\neg \exists x P(x)$. Let x be arbitrary. Suppose $P(x)$. Since we have a specific $x$ for which P(x) is true, it follows that $\exists x P(x)$, which is a contradiction. Therefore $\neg P(x)$. Since $x$ was arbitrary, we can conclude that $\forall x \neg P(x)$, so $\neg \exists x P(x)  \implies \forall x \neg P(x)$    What makes me confused is the line in bold text, that is "Suppose $P(x)$". The reason is that he writes that he is doing proof by contradiction, but $P(x) \not\iff \neg\forall x \neg P(x)$. He is not treating all possible cases where $\forall x \neg P(x)$ is false by assuming $P(x)$, does he? So how can this be a valid proof by contradiction?    Edit: I deleted my first example since it was clearly not representative for the problem.  Edit 2: So the thing that I am opposing I guess is that $\neg \forall x \neg P(x)$ is not $P(x)$. And from what I have undestood, a proof by contradiction should use the negation of the consequent which I do not think that Velleman does. Convergence/Divergence of infinite series $\sum_{n=1}^{\infty} \frac{(\sin n+2)^n}{n3^n}$ $$ \sum_{n=1}^{\infty} \frac{(\sin n+2)^n}{n3^n}$$  Does it converge or diverge?   Can we have a rigorous proof that is not probabilistic?   For reference, this question is supposedly a mix of real analysis and calculus. I've used Windows for many years, using Dreamweaver as my editor as it had a file tree which is based on a project (via Local, FTP or SFTP).  I need a web development IDE with the following functionality:   syntax highlighting for PHP, HTML, CSS, JavaScript  file tree - Local, FTP, SFTP, Project-Based (e.g. via the methods of Local/FTP/SFTP) in other words, like Dreamweaver can open multiple files (tabbed)   Additionally, I wouldn't mind features like debugging, class listings, etc, as long as I can hide them away. Google App Engine: Is it possible to do a Gql LIKE query? I'm a teacher and will ask my students to either bring their laptops with Ubuntu installed with an specific list of packages preinstalled or bring a USB Drive or External Hard Drive with Ubuntu and the packages preinstalled.  For those who are going to bring the USB are two posibilities:   Install with the Ubuntu's "Startup disk creator" program, and telling it to reserve some space for persistence. Boot with a LiveCD and Install Ubuntu in the USB drive.   Then, in either case, boot with that media and install the packages.  So, for my purpose or any other purpose:   What is the best choice and why? Do we have another option? full explanation please.