$(*)$ Let $\{ x_n \}$ be a sequence where $x_n$ is either $1$ or $-1$. Then   $$ \sum_{n=0}^{\infty} \frac{ x_n}{n!} \; \; \; \text{is irrational}$$  This problem arise when I was trying to prove that $e$ is irrational. If the result above is true, then we can put $x_n = (-1)^n $ and hence   $$ \frac{1}{e} = \sum \frac{(-1)^n}{n!} \; \; \; \text{is irrational} $$  which would imply that $e$ is irrational.   So, my question is: Is $(*)$ true?
How can one prove that the series $\displaystyle \sum\limits_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number? There's no need to use Taylor expansion, integrals or any advanced/professional techniques. It can be proved using only basic techniques. My first attempt was to assume that the series does converge to a rational number $ \frac{p}{q}$ and then it follows by definition that for every positive epsilon there exists an integer k such that $ {\sum_{n=1}^{k} \frac{1}{(n!)^2} -\frac{p}{q}}&lt;=\epsilon$ I tried to get a contradiction but I failed.
Iterating over Java collections in Scala I'm writing some Scala code which uses the  API. I would like to iterate over the rows contained in the java.util.Iterator that I get from the Sheet class. I would like to use the iterator in a for each style loop, so I have been trying to convert it to a native Scala collection but will no luck.  I have looked at the Scala wrapper classes/traits, but I can not see how to use them correctly. How do I iterate over a Java collection in Scala without using the verbose while(hasNext()) getNext() style of loop?  Here's the code I wrote based on the correct answer:  class IteratorWrapper[A](iter:java.util.Iterator[A]) {     def foreach(f: A =&gt; Unit): Unit = {         while(iter.hasNext){           f(iter.next)         }     } }  object SpreadsheetParser extends Application {     implicit def iteratorToWrapper[T](iter:java.util.Iterator[T]):IteratorWrapper[T] = new IteratorWrapper[T](iter)      override def main(args:Array[String]):Unit =     {         val ios = new FileInputStream("assets/data.xls")         val workbook = new HSSFWorkbook(ios)         var sheet = workbook.getSheetAt(0)         var rows = sheet.rowIterator()          for (val row &lt;- rows){             println(row)         }     } } Blaise Pascal's famous wager was that even if the existence of God cannot be determined through reason, a rational person should wager as though God exists, because living life accordingly has everything to gain, and nothing to lose.  What logical flaws and/or fallacies (if any) are committed in making this argument? Installing Wi-Fi driver for Realtek Semiconductor RTL8723DE Device [10ec:d723] How do I install Wi-Fi drivers for Realtek RTL8723DE device in Ubuntu 16.04?  lspci -v | grep -i network 02:00.0 Network controller: Realtek Semiconductor Co., Ltd. Device d723 Add a duplicate:0|1 search operator Is there a way to calculate a sorted field with sequential numbers?  I have seen  that outlines how to calculate sequential numbers, but this is always calculated on FID order, not on sorted order.  #Pre-logic Script Code: rec=0 def autoIncrement():      global rec      pStart = 1       pInterval = 1      if (rec == 0):           rec = pStart       else:           rec += pInterval       return rec  #Expression: autoIncrement()   An example of what I'm trying to do.  I've used an advanced sort to sort by year, month, day, and now want to have sequential numbers in the Seq field.  You'll see that my OBJECTID field is not in order, so the above code won't work.    Can this be done either in the Field Calculator or using an Update Cursor in arcpy? Is it safe to cancel CHKDSK when it has been started without any parameters? I know that it is not a good idea to forcibly stop a chkdsk process that has been executed with parameters like /f and /r.   But is it safe to stop the process when it has been started simply by the chkdskcommand? Chkdsk prints out a lot of errors and is continuing very slowly.  I suppose it to should be safe to cancel the process since there should be only in read mode, correct? When transiting Moscow (SVO), citizens of which countries need a transit visa? How to burn or mount an ISO file? How can all labels of a given type of node be restyled in Tikz?  Imagine that there are nodes of different styles, and that only the nodes of style X should have red labels. Is this possible without having to change all labels one by one of nodes of style X? Word for an event that will likely never happen again I've read , but none of them are what I am looking for. I once read a book that used a word to describe an event that has happened in the past, but the odds were so astronomical that it will likely never happen a second time. The context I read these words in were the creation of the first life on earth (without any theological ideas). Basically, the text said that the scenario for the creation of the first living organism happened on a planet where conditions were just right by complete chance, and that the odds of this were so astronomical that it would likely never happen again anywhere in the universe over any length of time.  Sample sentence: "The creation of the first organisms was incredibly unlikely, the event can be called a _________ event, or an event that will likely never happen again."   I can't pretend to understand the validity of the claims by the text, but the words they used to describe these events were quite clever (and scientific sounding) that it would make a fun hyperbole.  I thought it was something along the lines of singularity event, but that doesn't sound quite right. Radially symmetric charge distribution (dipole moment) a) There's a radially symmetric charge density $\rho(r)$ centered around the origin. Determine the dipolemoment of that charge density. b) Let $\rho(r)$ be an arbitrary charge density now. Under what circumstances does the dipole moment of the displaced charge density  $\rho '(\vec{r}) = \rho (\vec{r}-\vec{b})$ differ from the one not displaced at all.  Here were my ideas so far: a) Just thinking about the situation it has to be zero, right? I mean, since there's no real dipole. But how do I show that mathematically? I was thinking of just going like this (it may be wrong): Let the charge density be $\rho (r)=kr$, then we can get the charge q by integrating: $$q=4\pi \int_0^R kr\cdot r^2dr=\pi k R^4$$ I'm looking at the charge distribution as a spherical electron cloud with radius $R$. Then, since $p=qd$ and $d$ is zero because there are no two different charges the dipole moment is zero. Is that sufficient as an answer? b) I don't know how to approach this one. My guess is that its dipole moment is also zero because we're only looking at a displacement here. Anyone got any idea? I would appreciate any advice on this. How to make git ignore changes in case? I'm not too sure what is going on here, but sometimes a particular file in my repository will change the case of its name. e.g.,: before: File.h after: file.h I don't really care why this is happening, but this causes git to think it is a new file, and then I have to go and change the file name back. Can you just make git ignore case changes? [edit] I suspect it is Visual Studio doing something weird with that particular file, because it seems to happen most often when I open and save it after changes. I don't have any way to fix bugs in VS however, but git should be a bit more capable I hope. According to Wikipedia      extra disks should not be mounted in /mnt/ according to the first link, but doesn't seams so according to the second link.  Both say thought that they shouldn't be in /media.  Question  So where should extra permanent disks be mounted? How to create an interactive bash script to collect arguments for a command in linux? I'm trying to create an interactive bash script where I can collect user input variables to be used as arguments in another command. Can someone provide me an example of this?  Currently i want to do:   what type of content you want to delete? what is the status?   then feed it into a mysql delete query command. How to execute a JavaScript function when I have its name as a string Localization of ideals at all primes Windows 8 hibernation option missing from power options Windows 8 Professional 32-bit running on Laptop. And the Hibernation option is missing from the Power Options in the System Settings. Even after I enable hidden power features by giving admin privileges.  How can I fix this?    Also the option to enable hibernation is missing from the advanced power plan options.      SOLUTION UPDATE:  Thanks to  I have found that you need to enable hibernation using a command prompt.   Get to the Start menu, type 'CMD' this will begin the search. Press the CTRL+SHIFT+ENTER to start the command prompt with administrative privileges. Type powercfg -h on and press enter I was reading this article about S&amp;P 500 short ETFs:     and I was confused when it said: "investors should note that such products are suitable only for short-term traders as these are rebalanced on a daily basis". What is ETF rebalancing exactly? and why does it make these ETFs unsuitable for long-term short trading? Square roots -- positive and negative Why doesn't this perpetual motion machine using the buoyant force work? I realize this isn't possible, but I can't see why not, especially if you change the model a little bit so that the balls simply travel through a tube of water on the way up, rather than exactly this model.  Please be clear and detailed.  I've heard explanations like "the balls wouldn't move" but that doesn't do it for me - I really do not see why the balls on the right would not be pulled/pushed up, and the rest of the chain wouldn't continue on.