Is the value of the sum $1-1+1-1+1-1+\cdots$ does not exist? Let $a_k:=(-1)^k$ where $k\in\mathbb{N}$. $\mathbb{N}$ is the set of all non-negative integer.  And we define the partial sum $S_n:=\sum \limits_{k=0}^{n}a_k$. Notice that the sequence $\{S_k\}$ diverges which also implies that the infinite series $\sum \limits_{k=0}^{\infty}a_k$ cannot be defined.  If I consider the infinite sum $1-1+1-1+1-1+\cdots$, then this statement is equivalent that I just defined the infinite series $\sum \limits_{k=0}^{\infty}a_k$, which is contradiction.  But, suppose that the sum $1-1+1-1+1-1+\cdots$ exists and let the value of the sum be $S$. Then, we can easily observe that $S=1-S$, therefore $S=1/2$. The supposition of this proposition already proved as false, but if I ignore the definition of infinite series, it holds.  Which one is right? Or are these propositions depends on what we define?
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!
How to clone a Windows 10 dell onto an SSD I've recently bought a Dell Inspiron 3650, and would like to clone the disk onto a samsung SSD I have bought for it, in order to speed up boot times etc.  Samsung provide a clone tool called "Samsung data migration", which I've had success with in the past for cloning a system from a HDD to SSD.  Having tried it on the dell, the disk is not bootable, Leaving me with an error that no bootable media can be found. I've tried using a recovery usb to fix any boot system errors, but this fails with an error.  Booting into windows on the old HDD, the system recognises the disks have the same ID and disables the SSD, so it looks like the clone was successful.  I've tried disabling secure boot in the bios and then doing the clone, but this doesn't make a difference.  What else should I try? The system has Windows 10 home installed and is brand new, straight out of the box. Capacitor with different charges on each plate I am confused as to whether/how capacitance changes when each plate has a different charge.  For example, consider a coaxial cable and put $20Q$ on the outer cable, and $-Q$ on the inner. Or how about concentric spheres, grounding either the inner or the outer? I use TeX Live 2010 and my editor is Texmaker Bidi. It was working probably good but from 2 months ago I've encountered with the error:   !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! Fatal fontspec error: "cannot-use-pdftex" !  ! The fontspec package requires either XeTeX or LuaTeX to function. !  ! You must change your typesetting engine to, e.g., "xelatex" or "lualatex" ! instead of plain "latex" or "pdflatex". !  ! See the fontspec documentation for further information. !  ! For immediate help type H &lt;return&gt;. !...............................................   when I'm trying to load xepersian LaTeX.  but I do not know how can I change pdfLaTeX to XeLaTeX.    This is the second edit of my question .   the code is :    ‎\documentclass{article}‎ ‎\usepackage{pstricks}‎ ‎\usepackage{‎pst‎-‎bezier}‎ ‎‎\usepackage{xepersian}‎  ‎\settextfont{XB Zar}‎  ‎\setlatintextfont{Junicode}‎  ‎\setdigitfont{XB Zar}‎  ‎\begin{document}‎  ‎!‎سلام دنیا‎‎ ‎\begin{pspicture}(‎3‎,3)‎ ‎\psbcurve(1,2)(5,7)(2,2)(12,3)‎ ‎\end{pspicture}‎ ‎ ‎\end{document}   but the output does not include the shapes . I've found several posts with this same question, but the answers always diverge to some other topic or they don't work for my needs.  After a node is saved, I want my custom module to insert a row into one of my custom tables which contains the nid of the new node. When hook_node_submit is called, the nid doesn't exist yet. hook_node_insert does have the nid, but only gets called the first time the node is saved, but I potentially need to take action each time the node is saved.   My plan is to separate my functionality into its own method and implement both hooks, but I'm trying to learn to do things properly with Drupal and so I'm wondering if there is a hook that actually fits with my requirements.  Is there a hook which is called every time a node is saved and called AFTER the save is complete? Where are Upstart log messages on Ubuntu 13.X? present continous or present perfect continous? Should I write 'I have been reading a book now' or 'I am reading a book now'? I started to read one week ago and I haven't ended yet. Thanks for help I'm looking for a word that means     someone (especially a boss) who is not pleased (with your work, attitude etc.) no matter how hard you try. php/mysql with multiple queries error while loading shared libraries: libssl.so.1.1 (and libcrypto.so.1.1) As you can see I'm missing libcrypto and libssl files. I locked out myself from my server after running apt-update and apt-upgrade, I'm running this from web console, because ssh nor any other program won't work because of missing SSL files.  I tried removing and installing apt install openssl so did libssl-dev and/or compiling own openssl from a distro. None of these create the needed .so files, I ran locate, those files are not present anywhere on the server. There's also libssl1.0.0 which cannot be reinstalled and removing it would cause removal of 700MB of packages that would literally kill server and delete all data. Is redirecting homepage to a page with keyword it's targeting helps in SEO? "Ghost" UI elements Occasionally, I get part of an application lingering behind on my desktop:    This is part of a context menu for an application. It is always top-most, meaning that it is always on the screen, and I cannot click on it (it just clicks through it). It has happened before, sometimes with whole windows. It remains on screen even if the application it originated from is no longer running.  How do I get rid of it without rebooting? Starting services in batch file on windows 7 What is the meaning of "algebraically indistinguishable" I heard the term couple of times (in Field theory class and book), for example: The different roots of $p(x)=x^3-2$ are "algebraically indistinguishable".  I understand the meaning intuitively, but what does it really mean ? How do I model the fighter's Great Weapon Fighting fighting style in Anydice? I was trying to create an AnyDice function to model the Great Weapon Fighting fighting style (which lets you reroll 1s and 2s), but I couldn't get it to work on any arbitrary dice.  I've found this one:  function: reroll R:n under N:n {    if R &lt; N { result: d12 } else {result: R} } output [reroll 1d12 under 3] named "greataxe weapon fighting"   And it works fine. But I don't know how to make the function generic so i don't need to change the d12 every time i want a different dice to reroll.  I've tried  function: reroll R:n under N:n {    if R &lt; N { result: d{1..R} } else {result: R} } output [reroll 1d12 under 3] named "greataxe weapon fighting"   but it is not giving the right probabilities. Maybe if I could fetch the die size inside the function... Using "recently" with past simple Is it fine to use recently with the past simple tense in the following sentence?     Did you read any books recently?   Is it grammatically correct?  Or do I have to use:     Have you read any books recently? How do I redirect command output to a file? Find probability using distribution function method Beamer how to change color of infolines and frame title How could I change the color of color of infolines and frame title in beamer. I am using this now   \documentclass[xcolor=dvipsnames]{beamer} \usetheme{CambridgeUS} \useinnertheme{rectangles} \useoutertheme{infolines} \usecolortheme[named=Brown]{structure}    I need to change the color from red to brown. [named=Brown] doesn't work for \useoutertheme{infolines}. This is my current output. I also need to change the background of frame title (mouse pointer) I don't know why but now, when i go in material preview or rendered (in solid mode it works fine), my object is entirely gray, or invisible idk, because it has the same color of the background. I've been working for days on this project without having this issue, it suddenly showed up. I tried opening other old projects and suddenly they all do the same so i guess it is some blender program setting that applies for all the projects, but idk how it happened cuz i didn't change anything :/, anyone got any tips? The functions of each pair are inverse to each other. For each pair, check that both compositions give the identity function.  $F: \mathbb{R} \to \mathbb{R}$ and $F^{−1}:\mathbb{R} \to \mathbb{R}$ are defined by $F(x)=3x+2$ and $F^{−1}(y)=\dfrac{y−2}{3}$. for all y ∈ R  My attempt:     Inverse Function  For each particular but arbitrarily chosen $y \in \mathbb{R}$, according to the definition of $f^{-1}$, $f^{-1}(y) = \dfrac{y-2}{3}$ is a unique real number $x$ such that $f(x) = y$. \begin{align*} F(x) &amp; = y\\ 3x + 2 &amp; = y\\ x &amp; = \frac{y-2}{3} \end{align*} Therefore, $f^{-1}(y) = \frac{y-2}{3}$.  Compositions of Functions  The functions $g \circ f$ and $f \circ g$ are defined as follows: $$(g \circ f)(x) = g(f(x)) = g(3x + 2) = 3x + 2$$  for all $x \in \mathbb{Z}$.