Show that $\lim_{n \to \infty} \prod_{k=1}^{k=n} (1+ \frac{1}{n}f(\frac{k}{n})) = e^{\int_{0}^{1}f(x) dx}$
Help finding the limit of $\lim_{n \to \infty}\prod_{k=1}^{n}\left(1+\frac{1}{n}f\left(\frac{k}{n}\right)\right)$.
Strange arrow mark with TikZ edge and anchors. When I draw an edge between two nodes with anchors and globally set option [-&gt;], I get an additional arrow tip symbol at the beginning of the last edge in the path. For example in  \documentclass{article} \usepackage{tikz}  \begin{document} \begin{tikzpicture}     \node (A) at (0,0) {$A$};     \node (B) at (1,0) {$B$};      \draw[-&gt;] (A.north east) edge (B.north west);      \draw (A.south east) edge[-&gt;] (B.south west); \end{tikzpicture} \end{document}   the first \draw command produces incorrect output, while the second produces correct output:    Am I doing something wrong, or is this a bug in TikZ? Do "twice as high xxxxx as" , "twice as high" and "twice higher" mean the same thing? I don't mean people with low self esteem.  But there are people who feel happy at being able to make other people laugh at their own expense. I remember someone telling me there is a word for them, I can't find it.  Let me explain:  I sometimes, while sitting with a group of people, might just tell them a personal embarrassing story, which might be completely fabricated, to make other people laugh. And when they laugh, I feel happiness.  I'm also self deprecating, but not in a way that I hate myself, rather,  just my work (I'm a screenwriter) and knowing I can do better. I do not self deprecate myself in public (I don't praise myself either).   Edit: Now that its mentioned, I think the word I'm talking about was used to describe a person's sense of humor. Or it could be a word used to describe a psychological trait/disorder.  Edit: I just looked up humor styles on Wikipedia. And after reading it, I guess the word that closely matches what I'm looking for is 'Affiliated Humor' but in a 'Self Defeating' style. Can I have too many mods? I found a lot of mods that make thinks lighter, and add mark and recall, and a basement I like. They are all on Steam, and what I want to know is if I can have too many? Will they all work together? Or should I just leave them alone? I installed Ubuntu on my Windows 7 Sager laptop using Wubi. Hibernate (i.e. suspend to disc) is not an option from the power icon, only suspend, shutdown, etc. Hibernate is also not an option from my battery/lid close options. I understand that hibernation is disabled by default in Ubuntu 12.04. I tried running pm-hibernate but I get the following message:  Looking for splash system... none  s2disk: Snapshotting system   and then the computer just hangs with a black screen. According to the documentation  if this fails then I can't enable hibernate but it offers no help in making pm-hibernate succeed.  Could swap be my problem? It looks like I have very small swap:  user@ubuntu:~$ cat /proc/swaps Filename                Type        Size    Used    Priority /host/ubuntu/disks/swap.disk            file        262140  0   -1   The advice on  is only for the author's set up (e.g. I don't have an Ubuntu install disk since I used Wubi) and he says that 'INFO: This will not work for 12.04, resume from hibernate work differently in 12.04.'  Any advice? I really need to get hibernate working to use my laptop as a, er, laptop.  Thanks How do I post code in chat? I would like to be able to post code in chat. I know that I can post inline snippets of code by using backticks, but I don't know how to post blocks of code, with the entire chat message formatted as code, and indentation respected. I am aware of . After reading the comments, I found out that I could format in "fixed width", but I can't see a "fixed width" button anywhere. What should I do? How do I redirect www to non-www in Route53? I host my site at domain.com.  My DNS entries in Route53 are as follows:  domain.com      A       xxx.xxx.xxx.xxx      300 domain.com      NS      stuff.awsdns-47.org  172800 domain.com      SOA     stuff.awsdns-47.org  900   I would like to redirect traffic from www.domain.com to domain.com, as currently this just returns a 404.  suggested a PTR record, and I added that:  www.domain.com  PTR     domain.com           300   but it didn't work. What should I be doing? Batch processing LiDAR files (.las to raster) with PyQGIS? for layers in glob.glob("*.las"):         processing.runalg("grass7:r.in.lidar", "input_las",5,1,"",1,100,"","",5,"","",False,True,True,"331289.57265,337990.42735,4558790.0,4560890.0",0,layers+"E:/las_new/.tif")      {'output': 'PNOA_2016_CAT_huso31_346-4560.lasE:/las_new/.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4562.lasE:/las_new/.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4564.lasE:/las_new/.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4566.lasE:/las_new/.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4568.lasE:/las_new/.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4570.lasE:/las_new/.tif'}   output_ras=("E:/las_Ras")   for layers in glob.glob("*.las"):     processing.runalg("grass7:r.in.lidar", "input_las",5,1,"",1,100,"","",5,"","",False,True,True,"331289.57265,337990.42735,4558790.0,4560890.0",0,layers+"output_ras.tif")      {'output': 'PNOA_2016_CAT_huso31_346-4560.lasoutput_ras.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4562.lasoutput_ras.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4564.lasoutput_ras.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4566.lasoutput_ras.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4568.lasoutput_ras.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4570.lasoutput_ras.tif'}     for layers in glob.glob("*.las"):     processing.runalg("grass7:r.in.lidar", "input_las",5,1,"",1,100,"","",5,"","",False,True,True,"331289.57265,337990.42735,4558790.0,4560890.0",0,"output_ras.tif")      {'output': 'output_ras.tif'}   {'output': 'output_ras.tif'}   {'output': 'output_ras.tif'}   {'output': 'output_ras.tif'}   {'output': 'output_ras.tif'}   {'output': 'output_ras.tif'}   I would like to batch process many LAS files to rasters using the PyQGIS algorithm. As you can see in the code, I think the final problem is with the output. It does not write the files out to my defined output pathway as tif files. Also how can I make sure the naming of output rasters are according to the original LAS they are generated from? What am I doing wrong here? Not a pro at Python. Started 3 days ago. $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers of $x$ and $y$ Give a convincing argument that $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers $x$ and $y$. Could someone please explain how to prove this? I attempted to say that the largest values that could be added to $x$ and $y$ is $0.99$ and that doing so still made $\lfloor x\rfloor + \lfloor y\rfloor &lt; \lfloor x+y\rfloor$. However, my answer was not accepted. How do I eliminate the vertical realignment when using arrayrulecolor with cmidrule? I'm using consecutive \cmidrule commands in conjunction with \arrayrulecolor to alternately hide and unhide horizontal rules in a table.  However, when \arrayrulecolor is used with \cmidrule, some odd vertical spacing is introduced. I'd like to eliminate it.  (I'm using longtable, so a colorbox, minipage, or anything else tied to a page won't work.)  Below is an image that introduces the challenge. The first block shows a \toprule.  My goal is to create a shaded border around the table, so I've added a dummy column.  The second rule shows a \cmidrule that skips the first column.  The third block shows how I can use a \cmidrule to fill in the whitespace with a shorter \cmidrule.  Finally, I want to recolor this short \cmidrule to the background color and restore the color later.  Note the jagged vertical skipping in case 4 compared to case 3.  Why would \arrayrulecolor change vertical alignment?  How can I eliminate the jagged edges?    Here is the code:   \documentclass[11pt]{article}  \usepackage{longtable} \usepackage{booktabs} \usepackage{color} \usepackage{xcolor} \usepackage{colortbl}  \setlength{\aboverulesep}{0pt} \setlength{\belowrulesep}{0pt} \setlength{\extrarowheight}{0.25ex}  \newcolumntype{L}{&gt;{\columncolor[gray]{0.8}}l} \newcolumntype{C}{&gt;{\columncolor[gray]{0.8}}c} \newcolumntype{R}{&gt;{\columncolor[gray]{0.8}}r}  \begin{document}  \textsf{ \begin{longtable}{LLLLL}  &amp; \multicolumn{3}{@{}L}{\small {\tablename\ \thetable{}} - demonstration of odd vertical alignment with arrayrulecolor} &amp; \\%  &amp; \multicolumn{3}{@{}L}{\small {case 1: the line below is a toprule}} &amp; \\% \toprule  &amp; \multicolumn{3}{@{}L}{\small {case 2: the line below is a cmidrule that spans the middle 3 columns.  Note the whitespace!}} &amp; \\%  &amp; \multicolumn{3}{@{}L}{\small {The challenge: hide the widespace giving the appearance of a spacing around a longtable.}} &amp; \\% \cmidrule[\heavyrulewidth]{2-4}  &amp; \multicolumn{3}{@{}L}{\small {case 3: the line below is 3 cmidrules, 1 each for the first and last columns and 1 for the middle}} &amp; \\%  &amp; \multicolumn{3}{@{}L}{\small {note how the cmidrules align vertically ... perfect!}} &amp; \\% \cmidrule[\heavyrulewidth]{1-1}\cmidrule[\heavyrulewidth]{2-4}\cmidrule[\heavyrulewidth]{5-5}  &amp; \multicolumn{3}{@{}L}{\small {case 4: Now, lets try recoloring the first and last cmidrules}} &amp; \\%  &amp; \multicolumn{3}{@{}L}{\small {how do I eliminate the odd vertical alignments?}} &amp; \\% \arrayrulecolor[gray]{0.80}\cmidrule[\heavyrulewidth]{1-1}\arrayrulecolor[gray]{0.0}\cmidrule[\heavyrulewidth]{2-4}\arrayrulecolor[gray]{0.80}\cmidrule[\heavyrulewidth]{5-5}\arrayrulecolor[gray]{0.00} \endfirsthead  &amp;  Arizona State University &amp;  &amp; ASU &amp;\\   &amp;   Baylor University &amp;  &amp; BAYLOR &amp;\\   &amp;   Boston University &amp; 164988 &amp; BU &amp;\\   &amp;   Brown University &amp; 217156 &amp; BROWN &amp;\\   &amp;   California Institute of Technology &amp; 110404 &amp; CALTECH &amp;\\   &amp;   Carnegie Mellon University &amp; 211440 &amp; CMU &amp;\\   &amp;   Case Western Reserve University &amp; 201645 &amp; CASE &amp;\\   &amp;   Columbia University &amp; 190150 &amp; COLUMBIA &amp;\\   &amp;   Cornell University &amp; 190415 &amp; CORNELL &amp;\\      &amp; Yale University &amp; 130794 &amp; YALE &amp; % \label{tab:schools} \end{longtable} }  \end{document} Can someone give me a short description and comparison of the different road cycling pedal / shoe systems? I am looking at buying a pedal/shoe system as the clips and straps I am using are hurting my feet again.  It's been a long time since I have bought a road pedal / shoe system, and I have forgotten the differences between SPDs, Look pedals and all the others.    Can someone give a short comparison of each pedal system with pros and cons of each? My usage is/will be mainly city street cycling and then road cycling (recreation, not racing).  I'd prefer if people would give the pros and cons of each and not their opinion on the "pedal system war".  :) However, "I use them because I find XYZ and it helps in ABC ways" is very useful. Regular Expressions: Is there an AND operator? Obviously, you can use the | (pipe?) to represent OR, but is there a way to represent AND as well?  Specifically, I'd like to match paragraphs of text that contain ALL of a certain phrase, but in no particular order. How to switch from Resident visa to Tourist visa status in the Schengen area? I have a 1-year residence permit from one Schengen country that is now expiring. Upon its expiration, I would like to get my 90/180 days validity as a tourist to stay, or return here as soon as possible.  I understand that the tourist short-stay scheme is based on the 90/180 days "counter". But what happens if my previous stay was not tourism but residence (as searching year). Wouldn't that mean that I can quickly exit and re-enter the Schengen area to activate my tourist stay?   Also, is there a way to change this purpose of stay without exiting the Schengen area? When searching on a site with  enabled (), I saw some  rendering issues: (and currently it's also in the )        Even though the LaTeX rendering is fine when opening the :        The LaTeX code is:  $f(n) = \begin{cases} 0,  &amp; \text{$ \omega&lt;0, \omega=1, \omega&gt;3 $} \\[2ex] -\omega + 1,  &amp; \text{$ 0\le\omega&lt;1 $} \\[2ex] \omega^2 - 1,  &amp; \text{$ 1&lt; \omega \le 2 $} \\[2ex] 3 \omega,  &amp; \text{$2 &lt; \omega \le 3 $} \end{cases}$ What should happen if a logo I made is similar to another? Note: This is a canonical post created so that we can close the many similar questions as duplicates as discussed in .   We have created a logo for a client and they love it. We've already started preparing the materials that will feature the new logo.   However, after we had created the logo, we found a logo for another company that looks eerily similar. We did not intentionally create the new logo to look like the one of the other company, in fact we didn't even know it existed beforehand, but it does regardless.  What should we do? Is this plagiarism? Can our client use the logo we have created? Should they? Or do we need to come up with another logo that's more different? I would like to know if my brother can fly by United Airlines from USA to El Salvador, and my father, from El Salvador to USA using the same round trip ticket, both of them have the same name. Semigroups and solutions of equation I have been playing around with Ubuntu installing new themes and and docks. Sometimes during the installation, whole screen just hangs. What is the best way in Ubuntu to Kill a process/application? Anything similar to Ctrl + Alt + Del of windows? Windows 10 upgrade led to grub rescue Oddly wide spaces in subsection title I have a problem, this is what it looks like:     And this is the code:  \subsection{Bakterienst\"amme und Anzuchtbedingungen}   Where does the huge whitespace come from?