There is a way to see that $\lim \frac{k}{\sqrt[k]{k!}}=e$ from these definitions?  $$e=\lim\left(1+\frac1k\right)^{\frac1k};\quad e=\sum\frac1{k!}$$  If it is not possible, how to deal with the limit of the title to get it limit? Thank you.
I'm trying to find  $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$   I tried couple of methods: Stolz, Squeeze, D'Alambert  Thanks!  Edit: I can't use Stirling.
Additive group contains exactly three elements of the set $\{p,p+q,pq,p^q,q^p\}$ How to install bicycle pedals without a notch on the spindle? I bought a pair of bicycle pedals that to my surprise had no notches on the pedal spindle to use when tightening them. I managed to screw them in by hand, but they are not quite tight since I can't use a spanner on them. So what would be the correct way to tighten them? Are there any special tools or similar that I'm supposed to use?  The offending pedals can be seen below. The spindle is, as you can see, completely smooth. Few days back I have installed ecryptfs, created a private directory using it. Now my Ubuntu login became super slow. I have tried removing ecryptfs from Synaptic. I found It, it can not be deleted that way. Somewhere I have read about this command  ecryptfs-setup-private --undo   But this command displays a rather confusing list of commands on terminal. I don't want to mess up my installation, neither want to loose my administrator account.  Now my question is, how can I safely remove /home/user/Private (.Private) and remove ecryptfs encryption ? My daughter, 13, has said she wants to be a boy. A little background: we moved to a new state a year and a half ago. In her new school, she felt isolated among her peers until she met a girl in art, and became good friends. She joined the drama club, and made  friends with some 8th graders. They joined “the diversity club”. This club is a LGBT club. The kids put on a play they wrote and starred in, which had homosexual connotations. “Romeo and Juliet”, but script  rewritten to include  2 females as the couple, for instance.   My daughter got so wrapped up in it, that her life was as if she was copying the play. She’s no longer friends with the friend from art, but remains close with the 8th grade crowd. They all labeled themselves. One week, my daughter said she was gay, then, changed to bi, then onto pan. (I had to google that one). Now she is emphatic about being disgusted with her body and wants to be a boy. She wants to donate all the “girly” clothes she picked out last year, and tossed out all the makeup she bought a few months ago.   All her life, she was the classic girly girl. She had crushes on boys, and she and this boy were inseparable since first grade. She was sad when it came time to move away. When we settled here, she became sullen and angry, putting herself down. She now sees a therapist who she really likes. We’ve been working on her self esteem issues. She’s wanting us to call her “he” and by a male name. I can’t do that. She’s my daughter. She is female. Had we seen this coming, maybe I could come to accepting. But it happened out of the blue.  Sometimes the old her comes back-I got a makeup sample and she was interested in trying it, looked at me, then quickly put it down and said “ewww, makeup”—just a few months ago, she bought a bunch. She’s now wanting to wear boys clothes and talks of wearing a suit to her prom. She is very impressionable, and seems affected by viral YouTube videos about kids coming coming out to their mothers, and the mothers gleefully embrace with happy tears. She has told me when she’s 18, she’s having her breasts removed.    Nothing seems normal to me anymore-in terms of how she always was-this new situation just doesn’t fit.  My gut tells me this is a phase.  I’m hoping it’s a phase. But with all I’m seeing in the media about this, I’m not sure. I love her; this is too much for me to process. I’m a stay at home mom; we were inseparable. I knew her. Until now.:(  She cut off all her beautiful hair, and wears boys shorts, baggy tee shirts, and bought this awful men’s deodorant spray. Please help me sort this out. I’m at my wits end.  Do you think this is a phase? Is this common? Technique for finding the centerline of a curved object? I'm curious to know if anyone knows of any effective techniques for finding the center line of a curved object.  In the screenshot below you'll see a crude version of what I mean.     Is there any way to get that line to perfectly trace the center, or is it easiest to just eyeball it?  Photoshop or Illustrator techniques are acceptable. Splitting file based on size, but make sure that it ends with newline I am able to use the split command successfully to split large file into multiple smaller files. This is being achieved by following command  split -b 1G $temp_path $final_filepath   But only caveat is that these files many times contain last line which is split across 2 files. Is there any way to avoid that using split or any other command ? Mirror changes to side of cube across all six sides Blank questions and answers I have white blank on the posts.    App Version: 1.0.89 Device Manufacturer: Meizu Device Model: m2   OS Version: 5.1 (1464689330)    I can't see the questions and the answers and the comments.  I saw that it's a common issue..  Does it have a fix?   EDIT -  I saw  answer. This answer is ~2 years old. It says that it will fix in the next update.  His version is 1.0.59 - mine is 1.0.89. Huge different.   The problem is still exists! Django - specify which model manager Django admin should use Are software titles italicized? The titles of freestanding works (books, movies, plays, albums, etc.) are italicized; smaller parts within these works (chapters in a book, articles in a magazine, songs within an album) are not (they are put in quotes, instead).  How are software titles treated?  My understanding is that if a particular program resembles a book or a dramatic work (like a video game), its title is italicized -- otherwise, it is treated as a proper noun, just like any other product.     I played Grand Theft Auto, instead of reading "Democracy in Ancient Greece" in Microsoft Encarta and preparing my presentation in PowerPoint.   Is this approach correct? "Seen" status is earlier than the user's last post I posted what I think is a clever answer to:    But then I noticed the question was posted a while ago on Mar 2.  I wanted to see what the chance of the user logging back in and seeing my answer, but the  says the user was seen on Feb 19.  I thought this was supposed to be the date they last logged in? How could they have asked a question on Mar 2 if they haven't logged in since February?  Thanks Bibliography with page numbers In GKP's Concrete Mathematics, there is a useful and interesting feature I would like to see in more books. The bibliography entries include:   a - the page numbers where the book or article is cited in the document. b - a few lines of review of the cited work.   It is a kind of mix between bibliography, index and review.  Are there stable LaTeX packages/BibTeX styles to do (a) and (b) ?  EDIT: I wrote (b) initially as "comments about the work cited". It could be understood as "gathering comments made in the book about the work and reproducing them in the bibliography".  This is not what I meant but this could be something useful for others. How can I delete an email address from Outlook's email history? I would like to know how to upgrade from my current version of Ubuntu to the next stable from the Command line interface? How to find $\lim_{n\to\infty}\frac{1!+2!+\cdots+n!}{n!}$? How can I make Firefox open magnet-links in Transmission? How to check file types of uploaded files in PHP? On the PHP website, the only real checking they suggest is using is_uploaded_file() or move_uploaded_file(), . Of course you usually don't want user's uploading any type of file, for a variety of reasons.  Because of this, I have often used some "strict" mime type checking. Of course this is very flawed because often mime types are wrong and users can't upload their file. It is also very easy to fake and/or change. And along with all of that, each browser and OS deals with them differently.  Another method is to check the extension, which of course is even easier to change than mime type.  If you only want images, using something like getimagesize() will work.  What about other types of files? PDFs, Word documents or Excel files? Or even text only files?  Edit: If you don't have  or  and system("file -bi $uploadedfile") gives you the wrong file type, what other options are there? Let $f(x)\in C^2:[0,1]\rightarrow\mathbb{R}$ satisfy $f(0)=0,f(1)=1,f'(0)=f'(1)=0$, prove that: $$\max_{x\in[0,1]}|f''(x)|\ge4$$  By using Taylor series I can prove that $\max_{x\in[0,1]}|f''(x)|\ge2$, but I cannot go further. How does gravity work on Star Wars ships? Inspired by the question , how does gravity work on the Millennium Falcon and other ships its size? The passengers are not strapped in and regularly move around just easily as they would on a planet with Earth's gravity.   The answer for the Death Star refers to gravity often being "coreward" (presumably taking advantage of the natural gravity an object of such substantial mass would have), but that doesn't apply for a small ship. So how is gravity accomplished? What is a rocket engine thrusting against in space?