So i was trying to evalue this limit: $$\lim_{n \to \infty}\frac{1}{\sqrt[n]{n!}}, n \in \mathbb{N}$$  This, of course, by common sense is equal to zero (since factorial grows a lot faster). Is there a way to prove this limit without having to tackle with proving function growth rate. I;m not sure how that would be done, but I believe i would have to expand the factorial function to set $\mathbb{R}$ in order to compare, and that's still beyond my abilities. Thanks.
How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
Tilde (~) File in Macintosh Directory Last night I discovered what looks like an unusual file named "~" (tilde) in the root of my "Macintosh HD" directory on my MacBook.    Finder reports the file is 2.2MB in size, of kind "TextEdit.app Document", and was created on 22nd January 2014 (so about a month ago).  I haven't noticed any operational issues with OS X or it's applications.   I haven't yet touched the file but would obviously like to remove it.  I am very conscious that '~' is a special alias to the user home directory and thus I would appreciate any recommendations in removing the file in a safe manner that isn't likely to nuke my home directory or screw up the operation of applications or OS X. I don't know whether to delete or rename and whether to use Finder or the command line.   It's entirely possible that I created the file by accident since I've recently been drafting and running various shell scripts that use the ~, user home alias.  MacBook Pro 15" (Retina, Mid 2012) Yosemite When do merchants start paying the actual value of items without bonus effects and perks? Is a DNA molecule a single strand of polynucleotide or two of them linked together? Our molecular biology teacher told us that a double helix of DNA was composed of two DNA molecules linked together by hydrogen bonds. The thing is, until now, I always thought a DNA molecule was composed of two strands, those being polynucleotides, both of them being linked together. I can't find a link which is saying the same as my teacher, even if it seems technically correct to call a double helix a dimer of two DNA molecules.  I was curious to know what was the exact terminology. How to make a good glow effect around a sun? I use  to store all articles I need and bibtex4word add-on (in MS Word) to maintain the reference list. So, I would like to know how to properly fill the "author" field in Jabref with multiple authors to appear them correct in the reference list.  For example, I've got the article with several authors:      E. Orti, J.L. Bredas, C. Clarisse   And I would like them to appear in this manner. But instead I get this:     C. C. E. Orti, J. L. Bredas I understand test -n &lt;expression&gt; to return false if the expression evaluates to a string length of greater than 0.  Why then does the following happen?  Macbook:~ echo ${#undeclared_variable} 0 Macbook:~ test -n $undeclared_variable &amp;&amp; echo Foo Foo   Bash 4.x  My expectation is that since test -n $undeclared_variable should evaluate false, that Foo should not be echoed.  If I use [[  ]] instead of test, it works as expected. Study path to Bayesian thinking? I am six years into a business role and have a bachelor's in physics and applied math/stats. Sean Carroll's (Caltech physicist)  opened me to the idea that Bayesian statistics is one useful way of thinking about anything - inevitably you hold a prior and you should update your credence as additional information becomes available.   Is there a path to training your intuition to thinking this way? Critically, it would require repeated practice with verifiable answers through either a course, or self study that includes many problems and solutions. I do not believe simply reading will do.  Possible resources, having read every related question on this site I could find:   "Probability Theory" by Jaynes. Pro: analytic; intuitive explanation of bayesian statistics. Con: prerequisites; missing problems/solutions. "Doing Bayesian Data Analysis" by Kruschke. Pro: includes problems &amp; solutions; requires only "algebra and rusty calculus". Con: works in R, which I think provides for less intuitive learning than the analytical (I may be wrong).   If it is a multi-year path I need to take, starting elsewhere, I am happy to do so! Ideally, I would avoid the frequentist methods, as I have no use for them.  My goal is not to be a scientist, but to leverage insight into how reality works to go above and beyond the established thinking in business.  Many thanks for any suggestions! Buffering GeoJSON file using Python? I'm trying to perform a simple buffer on GeoJSON file using Python script. I am able to do it on a shapefile using Python script and on GeoJSON using QGIS fix distance buffer tool but I can't figure out how to do it on GeoJSON using Python script. I read many posts on it but none of them include working solely on GeoJSON. I also know that it is possible to convert GeoJSON to .shp but I need to work only with GeoJSON files.  My code that works for .shp file:  from osgeo import ogr  ds = ogr.Open(r'C:\vector files\tryyy.shp', 1) in_lyr = ds.GetLayer() out_lyr = ds.CreateLayer('new_buff44',in_lyr.GetSpatialRef(),ogr.wkbPolygon) out_lyr.CreateFields(in_lyr.schema) out_defn = out_lyr.GetLayerDefn() out_feat = ogr.Feature(out_defn) bufferDist = -0.00015075  for in_feat in in_lyr:      geom = in_feat.geometry()     out_feat.SetGeometry(geom)     for in_feat in in_lyr:          geom = in_feat.geometry().Buffer(bufferDist)         out_feat.SetGeometry(geom)         for i in range(in_feat.GetFieldCount()):             value = in_feat.GetField(i)             out_feat.SetField(i, value)         out_lyr.CreateFeature(out_feat)  del ds print "finish"   When I change the file from shp to geojson in the org.Open line I get:      NoneType' object has no attribute 'CreateFields'   I thought that if I read vector file as OGR datasource object then it doesn't matter which format is the original vector file.   Can someone help me understand what I should change so that the code will work on GeoJSON files? I am a super newbie for Ubuntu and basically need Ubuntu for a Python package.  I would like to run a Python file in say e:/username/folder1/folder2/python.py. I tried every method online and the only reply is bash: cd./e: No such file or directory. I am not sure whether it is because of the bash windows or virtualenv.  When I use bash on Ubuntu on Windows, ls, it says VIRTUALENV_DIR. cd, it says /home/username. The cd .. etc changes the directory a little, but not another drive, nor the folder I would like it to find. It is well known that confidence intervals and testing statistical hypothesis are strongly related. My questions is focused on comparison of means for two groups based on a numerical variable. Let's assume that such hypothesis is tested using t-test. On the other side, one can compute confidence intervals for means of both groups. Is there any relation between overlapping of confidence intervals and the rejection of the null hypothesis that means are equal (in favor of the alternative that means differ - two-sided test)? For example, a test could reject the null hypothesis iff the confidence intervals do not overlap. Is there any rule for differentiating between the endings "th" and "ht"? Some words end in th (length, width), and others end in ht (height, fight, tonight, caught).    I sometimes have difficulties in spelling such words because I don't know which ending to choose. Is there any rule or pattern (even localized) in choosing one or the other? More options when flagging for migration Show $d_{1}(x_{n},x)\rightarrow{0}$ if and only if $d_{2}(x_{n},x)\rightarrow{0}$. Let $X=(0,\infty)$. Define two metrics on $X$ by   $d_{1}(x,y)=|x-y|$ and   $d_{2}(x,y)=|x-y|+|\frac{1}{x}-\frac{1}{y}|$   for all $x, y \in{X}$.   Let $(x_{n})$ be a sequence in $X$ and $x\in{X}$.   Show  $d_{1}(x_{n},x)\rightarrow{0}$ if and only if $d_{2}(x_{n},x)\rightarrow{0}$.  Note: $d_{1}$ and $d_{2}$ are then said to be equivalent metrics on $X$.  Since $d_{1}$ and $d_{2}$ are equivalent metrics, will it be sufficient to show $c*d_{2}(x,y)\leq{d_{1}(x,y)}\leq{C*d_{2}(x,y)}$?  So far I have the following: $d_{1}(x,y)=|x-y|\leq{|x-y|+|\frac{1}{x}-\frac{1}{y}|}=d_{2}(x,y)$. Therefore, $d_{1}(x,y)\leq{C*d_{2}(x,y)}$ where $C=1$. Is there a $c$ that would satisfy the left-hand side of the inequality?  Any help would be appreciated. (this is not homework) Probability that a k digit number does not contain certain digits I am wondering if my proof is correct? Thank you for whoever willing to take a look at it for me.     Proof $\liminf E_k \subset \limsup E_k $   If $\{E_k\}_{k=1}^\infty$ is a sequence of sets, we define \begin{align*} \limsup E_k &amp; = \bigcap_{j=1}^\infty\left(\bigcup_{k=j}^\infty E_k\right)\\ &amp;= \bigcap_{j=1}^\infty(E_j \cup E_{j+1} \cup \cdots)\\ &amp;= (E_1 \cup E_2 \cup \cdots) \cap (E_2 \cup E_3 \cup \cdots) \cap \cdots. \end{align*}  Therefore, $\limsup E_k$ consists of those points in $\mathbb{R}^n$ which belong to infinitely many $E_k$. \begin{align*} \liminf E_k &amp; = \bigcup_{j=1}^\infty\left(\bigcap_{k=j}^\infty E_k\right)\\ &amp;= \bigcup_{j=1}^\infty(E_j \cap E_{j+1} \cap \cdots)\\ &amp;= (E_1 \cap E_2 \cap \cdots) \cup (E_2 \cap E_3 \cap \cdots) \cup \cdots. \end{align*} Therefore, $\liminf E_k$ consists of those points in $\mathbb{R}^n$ which belong to all $E_k$ for $k \geq k_0$.  Now consider $x \in \liminf E_k$, then $x \in E_1 \cap E_2 \cap \cdots$, or $x \in E_2 \cap E_3 \cap \cdots$ and so on, that is to say, $x \in E_{k_0} \cup E_{k_0+1} \cup \dots$ for some $k_0$. More specifically, $x \in E_k \forall k \geq k_0$. Therefore, $x \in (E_1 \cup E_2 \cup \cdots) \cap (E_2 \cup E_3 \cup \cdots) \cap \cdots$, so we showed $x \in \limsup E_k$. Book, it follows a guy in a dystopian future How do I get my arrow keys working in the at> prompt Force new users to go through animated tutorial before they post for the first time Do not allow people to post the minute they created an account.  Let them ask the question but not allow them to post.  Let them stew for 30 minutes and force them through a nice animated tutorial on what they should do before they post and enforcing the fact that bad questions will not get an answer.  If they had a Need to know! basic rules introduction forced on them asking them at the end to check a box I understand! some might ask better questions and help the community instead of flooding the site with bad questions.  The presentation should be on a timer without the ability to skip and highlight the similar questions asked before that might give them the answer.  Also for the first next 9 posts they should have to check a box next to the first 5 of each similar questions before posting.  I do not mean to be harsh, but the truth is I see more negative votes on questions then positive. It's like 90% of the time people give -1 to first questions. Font corruption: lines through characters I have a problem where a one-pixel high white line is displayed through certain letters of text. Almost always only a single character type is affected at once, and only in one font size (though that size can change over time). Most of the time things are fine, but this happens several times a day. It happens frequently in Firefox, but system dialogs, menus, etc. can also be affected. It's been happening ever since I upgraded to 10.10. Only restarting seems to fix it.  Two examples (in the first, only the larger "r" is corrupted):     Any help would be appreciated, thanks.  Edit: I've since noticed that I can (sometimes) make the fonts appear correctly by zooming in/out all the way and returning to the default in Firefox (one or two steps, then going back, does nothing) seems to make the problem go away temporarily.  Edit 2: I figure it's worth noting that the solution below doesn't fully work for me any longer (I'm now using Ubuntu 11.10). It does significantly reduce the frequency of this problem occurring.  Edit 3: Still around in 12.04. The fix provided by Vanni works perfectly now for me, but apparently doesn't for some.  Edit 4: This seems to be an xorg problem, with Vanni's answer fixing it for many Intel video cards. I've had this occasionally happen with AMD graphics cards in 13.10 and 14.04, but very rarely (only two or three times). Unable to send mail to hotmail from rackspace cloud