How to prove $(\frac{n+1}{n})^n<n $
Math Induction Proof: $(1+\frac1n)^n < n$
self referential struct definition? I haven't been writing C for very long, and so I'm not sure about how I should go about doing these sorts of recursive things... I would like each cell to contain another cell, but I get an error along the lines of "field 'child' has incomplete type". What's up?  typedef struct Cell {   int isParent;   Cell child; } Cell; Block any query in BIND How to make 'python' program command execute Python 3? The python program command executes Python 2. Python 3 can be executed using the python3 command.  How can Python 3 be executed using the python command? In Einstein's General Relativity, do the space-time dimensions curve? In Einstein's General Relativity, do the space-time dimensions curve according to the positions of stars, planets, and masses? Checking my understanding: $1 - 1 + 1 - 1 + 1 - ... = \frac{1}{2}$ I've recently run into a proof that claims that $1 - 1 + 1 - 1 + 1 - 1 ... = \frac{1}{2}$ that proceeds as follows:     Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$. Then $$S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ...) = 1 - S$$ Therefore, $2S = 1$, so $S = \frac{1}{2}$. QED   I was under the impression that this sum doesn't converge, and therefore this step of the proof is invalid:     Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$   That is, it's not even valid to suppose that the sum is equal to some (implicitly real or complex) number $S$, and therefore the reasoning that follows is meaningless because $S$ doesn't exist and therefore has any properties we want it to have.  Is the reasoning I've given above correct? That is, is it correct for me to claim that the proof fails because $S$ doesn't exist?  (I've heard that there are techniques for evaluating divergent integrals by using complex analysis and Taylor series, so perhaps there's another way to prove that the summation is $\frac{1}{2}$; I just wanted to see whether my reasoning is sufficient to explain why this particular proof is incorrect. If the proof actually is correct, then I stand corrected!)  Thanks! Complement property of DES I'm supposed to show the complement property of DES ($c=DES(p,k) =&gt; \bar c = DES(\bar p, \bar k)$).  My idea was to just start the algorithm with $\bar p$ and see what happens (ignoring the initial permutation for now). I looked at it step by step and got the the point where $R1 = \bar L1 \oplus f(\bar k1, \bar R0)$. So I figured I had to show that $f(\bar k1, \bar R0) = \bar f(k1,R0)$. Is that even right so far? Because I got stuck at the point where the S-Boxes are used. Why would $S(\bar x) = \bar S(x)$?  Somewhere else I read something about a formal prove using induction, but I don't know how I would do that either... Where did the convention of using single dashes for letters and doubles dashes for words come from and why is continued to be used?  For example if I type in ls --help, you see:    -a, --all                  do not ignore entries starting with .   -A, --almost-all           do not list implied . and ..       --author               with -l, print the author of each file   -b, --escape               print octal escapes for nongraphic characters       --block-size=SIZE      use SIZE-byte blocks   -B, --ignore-backups       do not list implied entries ending with ~ ...   I tried googling - and -- convention even with quotes with little success. How can I disable the "Welcome, Name!" banner notification for Game Center? What do the slashes mean in the character tables? I've been reading the PF core rulebook, but confused over some of the data in the Character tables, and not found anywhere in the book that explains it, so wondering if someone could explain it?  Taking the Monk class table for example:     Level | Base Attack Bonus | ..... | Flurry of Blows Bonus      1st       |       +0          |   ....    |        -1/-1      2nd       |       +1          |   ....    |        +0/+0      ...      ...      16th  |       +12/+7/+2   |   ....    |        +14/+14/+9/+9/+4/+4/-1      17th  |       +12/+7/+2   |   ....    |        +15/+15/+10/+10/+5/+5/+0      18th  |       +13/+8/+3   |   ....    |        +16/+16/+11/+10/+6/+6/+1   What do the slashes between the bonuses mean? Proof of Laplace transform of real number powers: $\mathcal{L} \{ t^n \} = \frac{n!}{s^{n + 1}}$ How to find all files with the same extension within a directory I want to search for all .conf files in /etc/. I tried grep -r *.conf /etc, but the result isn't right. What am I doing wrong? How to pull email address and password from text file with CMD? I have a text file named file.txt Which contains,  Combo: testing123mail@gmail.com:password As Combo:  abcdefghijklmnopqrstuvwxyz@gmail.com:ajfbdf some text here some more text here As Combo: hadvsfhia@ahfjbadj.com:password@1 some random text here iloveu@gmail.com:passypassyword123 goldenmammy@hotmail.com:youtube123   I am running the below code, to filter only mail:pass from the above text file file.txt  for /f "tokens=3" %%a in ('type file.txt^|find "As Combo:"') do &gt;&gt;mail.txt echo %%a for /f "tokens=2" %%b in ('type file.txt^|find "Combo:"') do &gt;&gt;mail.txt echo %%b for /f "tokens=1" %%c in ('type file.txt^|find ":"') do &gt;&gt;mail.txt echo %%c   Expected Output in mail.txt,  abcdefghijklmnopqrstuvwxyz@gmail.com:ajfbdf hadvsfhia@ahfjbadj.com:password@1 testing123mail@gmail.com:password iloveu@gmail.com:passypassyword123 goldenmammy@hotmail.com:youtube123   But, am getting actual output is,  abcdefghijklmnopqrstuvwxyz@gmail.com:ajfbdf hadvsfhia@ahfjbadj.com:password@1 testing123mail@gmail.com:password Combo: Combo: Combo: As As iloveu@gmail.com:passypassyword123 goldenmammy@hotmail.com:youtube123   Note: The mail id and passwords will be differ in different text file. Above given mail id and passwords are typed by me randomly for example.  Kindly fix my code and help to get expected output. Thanks in advance. Open a direct file on the hard drive from Firefox (file:///) I have a small PHP system I created for myself. This system contains a few .doc and .docx file documents. I want to be able to open them directly from the browser without downloading them.  I created a link: &lt;a href="file:///c:/mysite/myfile.docx"&gt;myfile&lt;/a&gt;. But Firefox does not open the link. href="http://localhost/myfile.docx" works fine - it offers me to download the file.  Any ideas on how I can do it? Do I need to change something in the Firefox options or security settings? Proving the second root of a quadratic equation If $\alpha$ is a root of the equation $4x^2+2x-1=0$, then prove that $4\alpha^3-3\alpha$ is the other root. How do I proceed? The sum of the roots, the product of the roots lead me nowhere. Should I find the roots of the equation and substitute in the given two expressions of $\alpha$ and check whether manipulating the first root gives me the second root (which seems much complicated), or is there any other way? Understanding the Basic Context of Trusted Hosts I got the trusted host settings error.      The trusted_host_patterns setting is not configured in settings.php. This can lead to security vulnerabilities. It is highly recommended that you configure this. See Protecting against HTTP HOST Header attacks for more information.    I read the  but I'm not understanding some of the basic context.   Are the "trusted hosts" me and other administrators who want to be able to administer the site remotely?  If so, would I add my PC's domain to the trusted host list? Let $p$ be a prime number. Let $R$ be the subset of all rational numbers $m / n$ such that $n \neq 0$ and $n$ is not divisible by $p$. Show that $R$ is a ring. My Proof-trying. $R1$. Associativity for under addition is clear. $R2$. $\dfrac {m} {n}+0=0+\dfrac {m} {n}$, so there exist the identitiy element for under addition. $R3$. $\dfrac {m} {n} + \dfrac {-m} {n}=\dfrac {-m} {n} +  \dfrac {m} {n}$, so there exist inverse element. $R4$. $\dfrac {m} {n} + \dfrac {m'} {n'}=\dfrac {m'} {n'}+\dfrac {m} {n}$, so commutativity. So $R$ is an additive abelian group. $R5$. $\dfrac {m} {n} ( \dfrac {r} {s} + \dfrac {t} {v})=\dfrac {mr} {ns} + \dfrac {mt} {nv}$ and $( \dfrac {r} {s} + \dfrac {t} {v})\dfrac {m} {n}=\dfrac {rm} {sn}+\dfrac {r} {s}+\dfrac {tm} {vn}$ $R6.$ $R$ is associativity for under multiplication, clear. $R7.$ $1 \dfrac {m} {n}=\dfrac {m} {n} 1 = \dfrac {m} {n}$, so there exist identity element for under multiplication. Can you check my answer? If false, can you write true answer? How can I prevent URLDownloadToFile from retrieving from the cache? I am using URLDownloadToFile to retrieve a file from a website. Subsequent calls return the original file rather than an updated version.  I assume it is retrieving a cached version. Is there a word for a comment which makes no sense or adds nothing to the current discussion? Escape string for use in Javascript regex How to check if current thread is not main thread