Reorder this series to change its sum
Evaluating the sum after reordering an infinite series $1-\frac {1} {2}+\frac {1} {3}-\frac {1} {4}+\ldots $
If the universe is expanding, what is it expanding into? If the universe is expanding, what is it expanding into? When the big bang happened where did it occur?  When the big bang happened how did it occur?  Where did the energy come from? Energy can not be created or destroyed does that mean, energy has existed before the universe was here? Is any particular punctuation necessary when a quotation is the subject of a sentence? For example, in the sentence below, should there be a comma after export? Or anything else?     "Laughter is America's most important export" is a well-known quote of Walt Disney's. So I tried a few different answers that I found here, and I feel that this is possibly a duplicate question, but everything I have tried from the other solutions hasn't worked.  Basically I am trying to bake a material to a texture that will be used in Unity, but every test gives me a black background image.  In the image below, the top image texture is the mask (black and white, white being the leaves), and the bottom image texture is the colour. I currently have an ec2 &quot;t2.nano&quot; instance up and running. I would like to switch to a &quot;t2.micro&quot;. In order to do that, I need to stop the instance. But I would lose my IPv4 address in the process. And I do not want that (because I have several ssh tunnels up and running from rasps in the wild that are currently hard coded to connect to this IPv4 address). I decided to create an Elastic IP address. But I understand that by creating an Elastic IP address and assigning it to my current ec2 instance, I would lose my current IPv4 address too. Is it possible to create an Elastic IP address from the current IPv4 address of my ec2 instance? If so, how can I do this operation? Of course when you type:  sudo apt-get install whatever   It asks you:   Do you want to continue [Y/n]?    Is there any way I can just install it without it asking me if I want to continue? I don't really care about space, I have more then enough. I am trying to solve:  $$\int_{-\infty}^{\infty}\frac{x\sin{x}}{x^2 + a^2}dx$$  and show that it equals $\pi e^{-a}$ for all $a &gt; 0$.  I translate it to its complex equivalent:  $$lim_{R \rightarrow \infty} \int_{-R}^{R}\frac{z\sin{z}}{z^2 + a^2}dz = \int_{C}\frac{z\sin{z}}{z^2 + a^2}dz - \int_{\gamma}\frac{z\sin{z}}{z^2 + a^2}dz$$  where C is a half circle with radius R, above the real axis, and $\gamma$ is only the curved (top part) of the half circle.  I've shown that $$\int_{\gamma}\frac{z\sin{z}}{z^2 + a^2}dz$$ tends to 0 as $R \rightarrow \infty$.  So now:  $$lim_{R \rightarrow \infty} \int_{-R}^{R}\frac{z\sin{z}}{z^2 + a^2}dz = \int_{C}\frac{z\sin{z}}{z^2 + a^2}dz$$  By residue theorem,   $$\int_{C}\frac{z\sin{z}}{z^2 + a^2}dz = 2\pi i \cdot res_{ai}f$$  as $ai$ is the pole of $f$ that lies above the real axis.  $$2\pi i \cdot res_{ai}f = 2\pi i \cdot lim_{z \rightarrow ai} (z-ai)\frac{z sin{z}}{z^2 + a^2}$$  By L'Hospital's rule, this is equivalent to:  $$2\pi i \cdot lim_{z \rightarrow ai} (z-ai)\frac{z^2\cos{z} + 2z\sin{z} - zai\cos{z} - ai\sin{z}}{2z} = 2\pi i \cdot \frac{\sin{ai}}{2} = \pi i \sin{ai}$$  But how is $\pi i \sin{ai}$ equivalent to $\pi e^{-a}$? Did I make a mistake? I feel uncomfortable saying sentences like the following:    "I've a car" instead of "I have a car" "They've a great time" instead of "They have a great time" "He's a pen" instead of "He has a pen" etc   I ask this because I read this sort of thing in a book.  Are they correct? And what is the rule? Can you use such forms in a formal setting? I am writing a WP plugin. I have created a button for the user to click in thw WP dashboard. I need to run a function when a user clicks the button, shown with a red arrow. The code snippet for the button is:&lt;p&gt;&lt;button class="button button-primary"&gt;Update Media Titles and ALT Text&lt;/button&gt;&lt;/p&gt;. I have already created the function in my class like so:  public function kh_update_media_seo() {      //update media files title and alt tags here }   I can handle the code that goes in the function alone, I only need help making the button in the WP dashboard fire off this specific function when clicked.  Pardon me if this sounds dump or straight forward. It's my first time doing this.  My plugin is a one file plugin if that helps. Why are low-rep users rate-limited when posting questions? I just encountered the following error:  Oops! Your question couldn't be submitted because: users with less than 125 reputation can only post questions every 20 minutes; try again later.  Why do you have this rule? Why 20 minutes? Bogus float inequality due to compiler optimization? Did spacetime start with the Big bang? Turn off autocomplete in PhpStorm Add leading zeroes to number in Java? How to confirm that mail has been delivered or not? I have 16GB of system RAM. On startup with no applications open except the task-manager Windows is using about 3gb of RAM. I looked in the processes tab, but nothing seems to be out of the ordinary. How can I find out why my Windows is using so much RAM.    all processes from all users      Reading from the poolmon it seems that my wireless broadcom driver is using about 0.4GB of of RAM. Even if I remove  it would still be using  2.6GB on start-up, which is still too much.      After reinstalling the wireless driver associated with the memory leak. I have a new screenshot and would like to confirm that it was indeed a memory leak. The situation is my  has been enchanted with .  The creature has      when it dies, return it to his owner’s hand   The enchantment says     When enchanted creature dies, return that card to the battlefield under your control with a +1/+1 counter on it.   Who gets the creature? Me or my rival?  Does the creature ability trigger before or after enchantment? Shell script works when saved with nano but not when saved with Notepad++ When I copy my bash script across from Notepad++ into a new file with nano editor inside SSH and save. It runs fine. (sh ./install).   But if I save the file (Exact same contents), upload to my web server, download it using Wget on the same machine. I get syntax errors. I've checked the encoding and they seem to be the same. I have since used a plethora of character encoding to see if it will resolve the issue. I am also setting the file to executable once I download it using wget!  The file runs fine and I have zero errors when copy and pasting using nano. Any idea what this may be? How can I monitor internet usage in my network? I need to know which computer is consuming my bandwidth. Is there a tool to do this? Underscores in words (text) How can I produce the text Word_one_two in LaTeX?  I tried:  Samp\_Dist\_Corr   But, it doesn't quite look right. Also, I want it in the typewriter font, so actually, I'm doing:  \texttt{Samp\_Dist\_Corr}   I find it looks a bit like the underscore is merging in to the bottom of the "D", but maybe it's just because of the typewriter D? let a,b be real  numbers and $\varepsilon&gt;0$. prove the above inequality. I dont know how to even start. please, I need your help.