As seen  and on this wikipedia page the sum of all the natural numbers to infinity is -1/12.    $\sum_{n=1}^\infty n = \frac{-1}{12}$  but the set of natural numbers is closed under addition and $\frac{-1}{12}$ is not a natural number.  In addition the series is clearly divergent, so how can we get away with "assigning" is a value as described on the wikipedia page.
$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) &gt; 1$.  Why should analytically continuing to $\zeta(-1)$ give the right answer?
Is there a travelling speed of for electric field? If yes, what is it? Consider two circuits containing a battery, bulb, switch and conducting wires but of length 1 meter in one case and 1000KM in other. When switched on at the same time both the bulbs glow instantaneously. I think it is because, electric filed travels from positive terminal of battery to negative terminal through wire almost instantaneously.  My question is there any speed for this electric field? If yes, what is it? By any modification in this circuit is it possible to induce any delay in glowing the bulb? What is the proper way not to scare civilians? Rigify generated rig, hand IK controller has unexpected rotation What does “would not have been possible had I remained” mean? What does this mean?      I am sure this would not have been possible had I remained a typical Anglophone North American.   I have never heard that before and I really find it strange.  I deduce that it means      I am sure this would not have been possible if I kept remaining Anglophone North American.   But I might be wrong in this. Distribution of Product of Random Variables with one being the normal distribution. Duplicate:    I want an pseudo random number generator that can generate numbers with no repeats in a random order. For example: random(10) might return 5, 9, 1, 4, 2, 8, 3, 7, 6, 10 Is there a better way to do it other than making the range of numbers and shuffling them about, or checking the generated list for repeats?  Edit: Also I want it to be efficient in generating big numbers without the entire range.  Edit: I see everyone suggesting shuffle algorithms. But if I want to generate large random number (1024 byte+) then that method would take alot more memory than if I just used a regular RNG and inserted into a Set until it was a specified length, right? Is there no better mathematical algorithm for this. I know questions on the "hot" tab of a site homepage are based on some sort of views/velocity criteria, but does anyone know the actual algorithm used?  Background: I was considering a feature-request about the "hot" tab, but I don't have enough information to speak about it intelligently. is piped ls the same as ls -1? ls returns output in several columns, whereas ls|cat returns byte-identical output with ls -1 for directories I've tried. Still I see ls -1 piped in answers, like ls -1|wc -l. Is there ever a reason to prefer ls -1? Why does ...|cat change the output of ls? How can I replace Windows with Ubuntu? Permission denied in 777 folder john@ubuntu01:~$ sudo ls -l /var/opt/gitlab/git-data/repositories/GL/www.git total 25068 drwxr-xr-x  2 git  git      4096 aug 14 19:58 branches -rw-r--r--  1 git  git        66 aug 14 19:58 config drwxrwxrwx  2 root root     4096 aug 15 14:56 custom_hooks -rw-r--r--  1 git  git        73 aug 14 19:58 description -rw-r--r--  1 git  git        23 aug 14 19:58 HEAD lrwxrwxrwx  1 git  git        47 aug 14 19:58 hooks -&gt; /opt/gitlab/embedded/service/gitlab-shell/hooks drwxr-xr-x  2 git  git      4096 aug 14 19:58 hooks.old.1471193907 -rw-r--r--  1 git  git      1329 aug 15 14:48 index drwxr-xr-x  2 git  git      4096 aug 15 13:58 info drwxr-xr-x 71 git  git      4096 aug 15 14:55 objects -rw-r--r--  1 git  git        98 aug 15 13:58 packed-refs -rw-r--r--  1 git  git  25618530 aug 15 14:03 post-receive.log drwxr-xr-x  5 git  git      4096 aug 14 20:31 refs john@ubuntu01:~$ ls -l /var/opt/gitlab/git-data/repositories/GL/www.git ls: cannot access '/var/opt/gitlab/git-data/repositories/GL/www.git': Permission denied   Here are ACLs  john@ubuntu01:~$ sudo lsattr /var/opt/gitlab/git-data/repositories/GL/www.git/custom_hooks -------------e-- /var/opt/gitlab/git-data/repositories/GL/www.git/custom_hooks/post-receive   I have never seen this before. No permission to LS even with 777? I need to share this folder for more convenient development. How can I solve it? Thank you.  EDITED: (here are your requests)  john@ubuntu01:~$ sudo ls -ld /var/opt/gitlab/git-data/repositories/GL/www.git drwxrwxrwx 8 git git 4096 aug 15 14:48 /var/opt/gitlab/git-data/repositories/GL/www.git john@ubuntu01:~$ sudo ls -ld /var/opt/gitlab/git-data/repositories/GL/www.git/custom_hooks drwxrwxrwx 2 root root 4096 aug 15 14:56 /var/opt/gitlab/git-data/repositories/GL/www.git/custom_hooks   EDITED MORE  sudo ls -ld /var/opt/gitlab/git-‌​data/repositories/GL/‌​www.git ls: cannot access '/var/opt/gitlab/git-‌​data/repositories/GL/‌​www.git': No such file or directory   hmmm... what is this? Gradient of $a^T X b$ with respect to $X$ How can I find the gradient of the term $a^TXb$ where $X$ is a $n \times m$ matrix, and $a$ and $b$ are column vectors.  Since the gradient is with respect to a matrix, it should be a matrix.  But I do not have a clue on how to derive this gradient.   Any help ? Install App Image such as beaker browser and pin it to dock (favorites) in Ubuntu 20.04? If $x\sqrt{1+y}+y\sqrt{1+x}=0$ find $y'$ I was speaking to an English learner and said, “All you have to do is read a lot.” And they thought that sentence wasn’t grammatically correct because I dropped the word to between is and read.  They thought it should be “All you have to do is to read a lot.” That sounds weird to me.   How do I explain to them the reason you drop the to? Or am I incorrect and it really should have a to? wireless network adapter supported on ubuntu? Execution of a command locally after ssh tunneling I have a script file ./run_app.sh which contains of the following:  ssh -L 8888:localhost:8888 ubuntu@address ./run_website.sh   # it should be executed locally   but it tries to run the second line on the remote machine. Actually, I want to run it on my pc. Any idea? We have to evaluate the following integral:  $$\int \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$$  I tried this:  I multiplied both the numerator and denominator by $\sec x$ And substituted $\tan x = t$.  But after that I got stuck.  The book where this is taken from gives the following as the answer: $$\ln(1+t)-\frac14\ln(1+t^4)+\frac1{2\sqrt2}\ln\frac{t^2-\sqrt2t+1}{t^2+\sqrt2t+1}-\frac12\tan^{-1}t^2+c$$ where $t=\sqrt{\cot x}$ Why isn't .profile sourced when opening a terminal? Just installed Ubuntu 14.04.1 LTS.  According to .profile,  # ~/.profile: executed by the command interpreter for login shells.  # This file is not read by bash, if ~/.bash_profile or ~/.bash_login exists.   There is no ~/.bash_profile or ~/.bash_login.  It does not seem to source .profile when opening a terminal.   Why doesn't it source .profile when opening a terminal? Why is .profile there if it is not sourced? How do I know/control what is sourced when a new terminal is opened? When a new terminal is opened, does it do a "login"? Where is a good place to add $HOME/bin to my path? .bashrc? Will that cause a problem because .profile has the same code? How to check if a service is running on Android? How do I check if a background service is running?  I want an Android activity that toggles the state of the service -- it lets me turn it on if it is off and off if it is on. Define Pressure at A point. Why is it a Scalar?