Looking for an algebraic proof of an unusual equality
Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction
How to modify the space between the numbers and text of sectioning titles in the table of contents? I'm preparing a document using the book class. In the table of contents, the space between the subsection number and title is too wide. How to reduce it? Is it OK to ask on SO for help in explaining someone else's working piece of code, from Open Source projects or library examples etc.? Or does this belong somewhere else? Replacement for alliums? Flag pop-up hidden beneath answer box (and it's still transparent!) The flag window on (a comment to) the last question is hidden beneath the answer box, as shown in the following screenshot:    Linked: , which is currently  - this is not fixed for flags on comments! How many ordered pairs $(A,B)$ they are if $A$ and $B$ are subsets of $\{ 1,2,3,4,5 \}$ such that $A \cap B = \{1\}$? How many ordered pairs $(A,B)$ they are if $A$ and $B$ are subsets of $\{ 1,2,3,4,5 \}$ such that $A \cap B = \{1\}$?  If 1 is in both sets :  Size of $A$ is $&gt;1$ and size of $B$ is $&gt;1$.  $\Rightarrow$ Size of $A \cup B =4$  $$5 \cdot (3 \cdot 3 \cdot 3 \cdot 3) = 405$$ is not correct.  What am I missing here? How to ensure that string interpolated into `sed` substitution escapes all metachars I have a script that reads a text stream and generates a file of sed commands that is later run with sed -f. The generated sed commands are like:  s/cid:image002\.gif@01CC3D46\.926E77E0/https:\/\/mysite.com\/files\/1922/g s/cid:image003\.gif@01CC3D46\.926E77E0/https:\/\/mysite.com\/files\/1923/g s/cid:image004\.jpg@01CC3D46\.926E77E0/https:\/\/mysite.com\/files\/1924/g   Assume the script which generates the sed commands is something like:  while read cid fileid do     cidpat="$(echo $cid | sed -e s/\\./\\\\./g)"     echo 's/'"$cidpat"'/https:\/\/mysite.com\/files\/'"$fileid"'/g' &gt;&gt; sedscr done   How can I improve the script to ensure all regex metacharacters in the cid string are escaped and interpolated properly? Not getting any reading from FTDI UMFT201XB USB-to-I2C module I got the  in order to interface with an I2C peripheral through a USB port on my Windows 7 computer.  The VCP drivers seem to install correctly, and a COM port is assigned to the module, and can be opened by a terminal emulator. However, I cannot seem to get any signal in or out of the module. That is, I see no activity when I send data through the port, or receive any data when I connect an I2C device to it (this was debugged using a scope). I have also verified that the RESET isn't on.  I found very little helpful information out there on how to get this module to work, so if anyone has experience with it, or with similar FTDI modules, my question is: what are the steps that I need to take in order to get the module to communicate? How to include audio file in PDF How do I get adequate depth of field in macro photography? I have been messing around with some Macro Photography using the techniques (and a reversal ring) described in a  by @ My main issue is that I am having a hard time getting the subject into focus. There are times when the center is blurred and the outside is in focus or the center in focus and the outside out of focus. What are ways to get as much of the subject in focus?   I am using a Nikon D5100, I have a 18-55mm kit lens but I find the 55-200mm lens easier to use for shooting. How to force the user to download a file, not to view it? Proof of Frullani's theorem How can I prove the Theorem of Frullani? I did not even know all the hypothesis that $f$ must satisfy, but I think that this are  Let $\,f:\left[ {0,\infty } \right) \to \mathbb R$ be a a continuously differentiable function such that $$ \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = 0, $$ and let $ a,b \in \left( {0,\infty } \right)$. Prove that $$ \int\limits_0^{\infty}  {\frac{{f\left( {ax} \right) - f\left( {bx} \right)}} {x}}dx  = f\left( 0 \right)\left[ {\ln \frac{b} {a}} \right] $$  If you know a more general version please give it to me )= I can´t prove it. My Mac Book Pro 13.-" doesn´t get past the grey booting screen with the apple and the spinning wheel (The loading bar appears but when it´s done nothing happens)   I have 220 GB in one single Partition (Macintosh HD) and there is all my work my music and the only copy of ALL MY PHOTOS. I´m so stupid I don´t have a backup of anything but know I have an external HD with 300GB space. I would like to repair my HDD but more important I need to save my files but I don´t know how to do it in Disk Utility Mode. I read there is a Diskwarrior but I don´t know how to install it without booting the mac. Please help!!!  When I start on Single User mode and enter /sbin/fsck -fs this appears  Executing fsck_hfs (version diskdev_cmds-557-393)  Checking Journaled HFS Plus volume.  The volume name is Macintosh HD  Checking extents overflow file.  Checking catalog file  Incorrect number of thread records  (4, 23745)  Checking multi-linked files.  Checking catalog hierarchy  Invalid volume directory count (It should be 175703 instead of 175701) Checking extended attributes file.  Incorrect number of extended attributes (It should be 340062 instead of 340064)  Checking volume bitmap.  Checking volume information.  The volume Macintosh HD could not be repaired after 3 attempts.  *****The volume was modified *****   What are the next steps if fsck fails to repair a drive? Longtable: page breaks within row possible? I am editing a longtable. Some of my cells contain a lot of text. Page breaks seem to happen only where table rows end/begin. I'd like, however, for the rows to span pages; it's fine with me if, for example, the first few lines of a row are at the end of one page and the rest of the row's lines are on the next page. Does installing many programs really slow down Windows? Why cast an unused function parameter value to void? $$\lim_{n \to\infty} \frac{n!}{n^n}$$  I have a question: is it valid to use Stirling's Formula to prove convergence of the sequence? Is it correct to sometimes capitalize words mid-sentence in computer user interfaces? Which LaTeX to install on Linux? Which LaTeX distribution would you recommend for using on Linux from command line? I am looking for one with most complete coverage of packages. When were the 'and' and 'or' alternative tokens introduced in C++? I've just read this  from Reddit.  They mention and and or being "Alternative Tokens" to &amp;&amp; and ||  I was really unaware of these until now. Of course, everybody knows about the , but and and or? Since when? Is this a recent addition to the standard?  I've just checked it with Visual C++ 2008 and it doesn't seem to recognize these as anything other than a syntax error. What's going on? Postfix Connection Timed out