I've simply said that adding $1$ to the numerator and denominator when the denominator is $5^n$ (greater than $2^n$), the &quot;$+1$&quot; accounts for a larger portion of the numerator than the denominator, so $\frac{2^n+1}{5^n+1}$ better be greater than$\frac{2^n}{5^n}$ for all $n\gt0$. I'm just not satisfied with this explanation because it seems sloppy and not rigorous... Does someone have a better proof?
Given a fraction:  $$\frac{a}{b}$$  I now add a number $n$ to both numerator and denominator in the following fashion:  $$\frac{a+n}{b+n}$$  The basic property is that the second fraction is suppose to closer to $1$ than the first one. My question is how can we prove that?  What I have tried:  I know $\frac{n}{n} = 1$ so now adding numbers $a$ and $b$ to it would actually "move it away" from $1$. But I cannot understand why $\frac{a}{b}$ is actually farther away from $1$ than $\frac{a+n}{b+n}$.  Why is that? What does it mean to add a number to both the numerator and denominator?
I use Chrome as my main browser.  I also spend a lot of time on a particular webapp that my company uses.  I'd like to keep those two things separate.  I can run the webapp in a separate window - but I'd like to go further than that.  I'd like to have it be identified by a different task in the Win 7 taskbar; ideally a different icon; and ideally a different title in Process Monitor.  How can I run a second, independent instance of Chrome, with a different title and icon, that won't merge with my main Chrome browser. Why is accuracy not the best measure for assessing classification models? What are the prerequisites for studying mathematical logic? I am looking to study mathematical logic, however, I find that introductory books are very daunting, which kind of disheartens me. You see, slowly but surely, I started to realize that the maths which I have learned did not just pop out of thin air, but is a collection of systems, which must of been developed via some other system, i.e, maths did not develop itself.  So I began to look into the origins of mathematics, and read that it was developed via a type of logic, which exists sort of by 'default', via a set of axioms, and then of course I looked up the definition of axioms.   So given that I'd be studying a type of logic whose origins are self evident axioms, naturally I believed there would be no prerequisites. However, in looking up mathematical logic, I have come across things such as Boolean algebra, sets, first order logic, some other type of logic, called 'traditional logic', as well as references to a sort of calculus, though not in a mathematical sense, I think.  So all in all, I am trying to develop a type of mental spider web, and I am trying to find out the strands which lye at the absolute bounds so that I may learn this mystique logic. Though I have no idea where to start.  Side note: This is the book I have started reading:   Credit goes to Wolfgang Rautenberg. Is there any way to bring an egg to its natural state (not boiled) after you cook it? This is probably a stupid question, but I'm curious now after I opened the refrigerator looking for eggs to make a cake and there was only one and it was cooked. Is there any way to undo that, whether at home or an industrial way? We should be able to find questions or answers that we have deleted I deleted one of my questions and would like to undelete it. This requires finding the question.  I think you shouldn't be prevented from seeing your own posts that you have deleted. It should be easier than me remembering the link or checking the browser history to easily look these up.   I would currently need 10K to see my deleted items and even then if I can find their URL. When something gets wet, it usually appears darker. This can be observed with soil, sand, cloth, paper, concrete, bricks...  What is the reason for this? How does water soaking into the material change its optical properties? $f(x)f(1/x)=f(x)+f(1/x)$ Find a function $f(x)$ such that:  $$f(x)f(1/x)=f(x)+f(1/x)$$  with $f(4)=65$.  I have tried to let $f(x)$ be a general polynomial: $$a_0+a_1x+a_2x^2+\ldots a_nx^n$$  which leaves $f(1/x)$ as:  $$a_0+a_1{1\over x}+a_2{1\over x^2}+\ldots + a_n{1\over x^n}$$  On comparing the coefficients of both sides, we see that:  $$2a_0=(a_0)^2+(a_1)^2+(a_2)^2+\ldots+(a_n)^2$$  And  $$a_1=(a_0a_1)+(a_1a_2)+ \ldots +(a_{n-1}a_n)$$  I don't know how to proceed further. I know I need to compare coefficients and come to a conclusion based on their values, but I don't see what to do next. Why are duplicates not being merged? I see more and more exact duplicates on SO. Instead of being merged they are being simply closed. I think this is very big disservice to SO. Why don't moderators merge duplicates? Why don't owners of the site do anything about this?  EDIT  It's disservice because this way we have more and more junk. Duplicates are duplicates and should be merged so that information in in one place.     If the duplicate has no answers then there is no benefit to merging.   Of course there is. You don't need to waste your time with dupes.     It also means that the search routes to the main question are   diminished which makes the posting of new duplicates more likely   There's easy solution for this which I gave in  back in march.     If both questions have answers that have been accepted, which one   should be unaccpeted?   There are plenty of solutions. You accept most voted accepted answer or the first accepted answer or both (I think this is be the best). Yes, there will be possibility of many accepted answers to one question. So what?     When a question is closed as a duplicate, there is a link to the   duplicate question. What more do you want?   I don't want yet another link. I want to have all answers to some problem in one place. Is this that hard? Isn't this natural expectaion?     Removal of the question means that the text in it is no longer searchable - that's a disservice.   See my answer above to Scrooge's comment. RELATED,ESTABLISHED state in iptables What do the words 'RELATED' and 'ESTABLISHED' individually signify in RELATED,ESTABLISHED state in iptables? How to move a file? In Avengers: Infinity War, up until Thanos      gets all six Stones   in order to use the Gauntlet he needs to close his fist, as we see in the fight scene on Titan.     Doctor Strange: "Don't let him close his fist!"   So why later does snapping work? Probability of success on third and fourth trials? I have a pretty basic probability question, but I'm just having difficulties remembering what distribution this is.  A coin is tossed until a head appears two times in a row. Given that we are using a fair coin, what is the probability that we toss the coin exactly 4 times such that the two consecutive heads are the 3rd and 4th trials?  I know that this would be fairly easy to solve by brute force and listing out the different probabilities, but I'm a little confused as to how to use a "shortcut" such as identifying this as a binomial distribution. I also don't know how to account for the probability that we are using 4 tosses.  Any help is appreciated. Thanks! Xcode free space requirement I am trying to update Xcode to 8.0. My OS is 10.11.6 El Capitan. However I am getting:     We could not complete your update   There is not enough disk space available to install the product.   I have got 14.22 GB free space. How much do I need?  UPDATE: I do not know the exact figure, though I did manage to update with 17.21 GB free space later on. Ok, so I have an index.php file which has to process many different file types.  how do I guess the filetype based on the REQUEST_URI.  If I request http://site/image.jpg, and all requests redirect through index.php, which looks like this  &lt;?php    include('/www/site'.$_SERVER['REQUEST_URI']); ?&gt;   How would I make that work correctly?  Should I test based on the extension of the file requested, or is there a way to get the filetype? Force update from unsigned repository I'm using an unsigned repo in Ubuntu 16.04 from Debian multimedia:  deb http://www.deb-multimedia.org jessie main   To install deb-multimedia-keyring, I'm running:  apt-get update &amp;&amp; apt-get install deb-multimedia-keyring -y   This gives an error:  W: GPG error: http://www.deb-multimedia.org jessie InRelease: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY 5C808C2B65558117 E: The repository 'http://www.deb-multimedia.org jessie InRelease' is not signed. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. Can I insert elements to the beginning of an element using .appendChild()? If the user selects an option in a dropdown box, there must be added a label and a textbox. Using appendChild, these elements get added to the end of the container.  var newFreeformLabel = document.createElement('label'); newFreeformLabel.innerHTML = 'Omschrijving:';  var newFreeformField = document.createElement('input'); newFreeformField.className = 'textfield'; newFreeformField.name = 'factuur_orderregel[]';  var newFreeformSpacer = document.createElement('div'); newFreeformSpacer.className = 'spacer';  container.appendChild(newFreeformLabel); container.appendChild(newFreeformField); container.appendChild(newFreeformSpacer);   The issue is that these elements should be inserted at the beginning of container, not at the end.  Is there a solution to this in PrototypeJS? Shouldn't the Terms of Service use Stack Overflow, Inc.? The  still uses the term Stack Exchange, Inc. However, I believe that the name is now Stack Overflow, Inc. , so I believe that should be changed to that. It is weird to have the wrong name in a legal document.   Yes, I actually read the Terms of Service. You should also do that. Create file and its parent directory I know that touch command creates a file:  touch test1.txt   but how I can create a file and its full path?  for example my desktop contains nothing:  ~/Desktop/$ ls ~/Desktop/$   and I want to create 1.txt in ~/Desktop/a/b/c/d/e/f/g/h/1.txt. Can I do this with a simple command like:  $ touch ~/Desktop/a/b/c/d/e/f/g/h/1.txt   instead of create full path manually and then create the file? Let $a,b,c\in \mathbb{Z}$. If $ac&gt;bc$ and $c&gt;0$, then $a&gt;b$   I think I would use contrapositive to prove this. But I am not sure proper contrapositive statement.   Is it If $a&lt;b$, then $ac&lt;bc$, and $c&lt;0$.? If so, contrapositive is not a good choice, isn't it? Find the probability of an event determined by an inequality on exponential random variables Let X and Y be independent random variables such that $X∼Exp(1)$ and $Y∼Exp(2)$.  Find the probability that $3X+4Y≤5$.  I thought the equation for solving this was the integral from $\int_0^a \int_ 0^{(a-y)} \lambda_1 e^{-\lambda_1 x}\lambda_2 e^{-\lambda_2 y} dx dy$ where $a = 5$, so I first started with the integral from $\int_0^5 \int_0^{(5-y)} 3 \times 1e^{-1x} \times 4 \times 2e^{-2y} dx dy$ and got $23.83828927$ which is much larger than a probability can be. Any help?