I need to prove or disprove:  I start with the Big O notation:  $\sum \limits_{i=1}^n $ log(i)  = Θ(n log n)  $\sum$ log i ≤ $\sum$ log n = n log n   (dont know if this is enough or not)
I'm interested in the exact barrier of $$\ln(n!)= \Theta(n\ln(n))$$  and if it even exists.  This means, there is a $c_1, c_2$ , so that $$\ln(n!) \le  \Theta(n\ln(n) c_1$$  $$\ln(n!) \ge  \Theta(n\ln(n) c_2$$  or  $$0 \le \lim\limits_{n \to \infty}\frac{\ln(n!)}{n\ln(n)} \ge \infty$$  I could use de l'Hospital for the limit fraction, since it's $\frac{\infty}{\infty}$ but the derivate of $\ln(n!)$ causes problems. Also I don't know how I could prove the two statements above or how to prove them wrong. Any help is appreciated.
Will my device work with Ubuntu? 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} My girlfriend is Russian. Last night we tried to fly her to the US on a valid tourist visa. She was stopped at DME and told she would be required to have $2700 USD to get back home immediately even though she had a return ticket. She said this was an "unwritten" rule because of the nonsense i keep seeing on CNN , Anybody else know anything? She is literally trapped and scared to death. I want to use the GameJolt API, how do I protect my private key which is in my game? GameJolt is a gaming platform/website that offers API access with features like achievements tracking for users, etc. I can use the API no problem on my tests but what I'm concerned about is the private key that is issued to my game. How do I ensure that it's protected from people that want to decompile my game? Once  they get the API private key, they will be able to do anything with it. Does walking fast prevent eggs hatching? The only thing I need is an exact copy of an existing shapefile in the same directory with a different name.  I have tried:   arcpy.CopyFeatures_management (ORLinks, MyLinks) #ORLinks: old links, MyLinks: new links   but for some reason, I cant make it a feature layer using:  arcpy.MakeFeatureLayer_management(MyLinks, 'MyLinks')   Error Encountered:  Error: Failed to execute. Parameters are not valid. ERROR 000732: Input Features: Dataset  M:\RAIL\Rail2.0.shp does not exist or is not supported Failed to execute (MakeFeatureLayer). How do I delete a folder that's in use? Sometimes when I'm working in Windows I'll get this prompt when I try to delete a directory:    Folder In Use The action can't be completed because the folder is open in another program Close the folder and try again.   However, the folder isn't "in use" at all, and I don't have it open. So how can I delete it short of rebooting? Isomorphism of $S^1$. (1)  If $E$ is a subset of $\mathbb R$ with finite outer measure, i.e. $m^{*}(E) &lt;\infty$; and   (2) $E$ is not Lebesgue measurable, i.e. there exists $F$ such that  $m^{*} (F) &lt; m^{*}(EF) + m^{*}(FE^{c}).$  [Claim] There exists an open set $O\supset E$ with finite outer measure, such  that   $$m^{*}(O E^{c})  &gt; m^{*} (O)  - m^{*} (E).$$  My  questions are the following.  (1) Is the above claim correct? I've seen this in Royden's book, but have a bit concern. Note that a set $E$ is Leb measurable if  it satisfies caratheodory condition: $m^{*} (F) = m^{*}(EF) + m^{*}(FE^{c})$ for  all subset $F$. If the claim is yes, it seems that measurability condition can  be reduced from all subset $F$ to Borel set $F$.  (2) If the claim is correct, a proof is needed. Show that $\frac{X-\mu}{\sigma}\sim N(0,1)$ using moment functions Let $X\sim N(\mu,\sigma^2)$. Show that $Z=\frac{X-\mu}{\sigma}\sim N(0,1)$ using moment generating functions. \begin{align*} M_Z(t)&amp;=M_{\frac{X-\mu}{\sigma}}(t)\\ &amp;=M_{X-\mu}\left(\frac t\sigma\right)\\ &amp;=e^{-\mu t}M_X\left(\frac t\sigma\right)\\ &amp;=e^{-\mu t}\cdot e^{\frac{t\mu}{\sigma}+\frac{t^2}{2}}\\ &amp;=e^{-\mu t+\frac{t\mu}{\sigma}+\frac{t^2}{2}} \end{align*} I'm not sure how to conclude that $Z\sim N(0,1)$. Coming from Windows I have gotten extremely used to the Windows Snap feature, which resizes a window when it's dragged to the screen edges.  However, it's not built into OS X. Are there any 3rd party utilities that add this functionality? How can I remove a commit on GitHub? Why are text format commands used inside braces? Most commands in Latex are written with the arguments inside braces following the command: \command{arguments}. But text formatting options, like \em, or \scriptsize are usually written inside the braces, like {\em some italicised text}. Why is this? What makes these commands different? I'm particularly asking because it seems to make standardisation - of which LaTeX really appears badly in need - more difficult. I found this odd relationship, $x^2 = \sum_\limits{k = 0}^{x-1} (2k + 1)$. How can I make Firefox download different file types to different locations? 98% of what I download is music, so I set the download location to my music folder. However, the other 2% has no place there, so every time I download .doc files or PDF files I have to remove them. Is there a way to make Firefox change the download folder based on file extension? I'll settle for all non-MP3 files getting saved to the desktop.  I know I can right click, choose Save Link As and pick the download location myself, but it seems like there should be a better way. UV mapping not filling faces How to use GROUP BY to concatenate strings in SQL Server? How do I get:  id       Name       Value 1          A          4 1          B          8 2          C          9   to  id          Column 1          A:4, B:8 2          C:9 How to set Google+ photos to be Public-Unlisted (as you could do with PicasaWeb)? Previously, if I uploaded photos from Picasa to PicasaWeb, I could specify them as Public but Unlisted. So you had to have the link to view them but no login was required.  I don't see that option now that Picasa uploads to Google+.  Any ideas? This error occurs whenever I try to work with sudo. I changed the permissions of /var/ folder to 777 and I think this causes the error. Please help me get back a functional sudo and /var. I'm trying to test a page that has an input but really its a drop down list. The user enters text and then a dropdown appears based on what the user inputted. They are groups that are in the database. If I look for these elements before typing anything they dont exist but once I start typing they do exist.  The problem is I cant just type the whole name and then press enter or return because an option has to be selected from the list.  I can press arrow down to get the focus on the item from the list but then if I say enter it doesnt select, it sends that enter to the original element and nothing gets selected.  I can find the select element:  var ddList = driver.FindElement(By.CssSelector(".ui-autocomplete-input")); ddList.SendKeys("Admins");   This is the first time the list appears in the html.  It looks like this:  &lt;li class="ui-menu-item" role="presentation"&gt;