How to prove that the sequence $n^{1/n}$ converges to 1
How to show that $\lim_{n \to +\infty} n^{\frac{1}{n}} = 1$?
It is common to see things described as the &quot;gold standard&quot;. It means that something is the best and is the ideal against which other things are to be compared; it cannot be superseded. I often , where a certain diagnostic test (e.g., the nasopharyngeal swab PCR test for infection with the Coronavirus causing COVID-19) is considered the most accurate and then other tests (e.g., cheaper or faster tests) are then evaluated against this benchmark. The etymology is covered in . In short, it is derived from the . Thus gold was the thing against which the value of all other currencies was measured. I don't like this term and would like to avoid using it. First, the desired meaning is not predominant; if you  you will mostly get results for the monetary system, not the more general phrase. Second, the etymology is outdated; the original meaning of the term now has the opposite connotation. The monetary gold standard was abandoned in the 1970s. Economists consider the gold standard to be a bad monetary system. Thus, if you are interested in economics and history, the term &quot;gold standard&quot; actually is associated with being outdated and flawed -- which is close to the opposite of the intended meaning. (Of course, many people are unaware of the original meaning of the term. ) (A note in response to comments: Why I want to avoid this term is relevant, but not essential, to the question. If you don't see a problem with the term, the question can be answered regardless.) My question: Is there a better phrase to use in place of &quot;gold standard&quot;? For example, what term could be used in a medical study comparing a new diagnostic test against the gold standard test? I'm not aware of any good substitute that has essentially the same meaning. Here are two example sentences where I would like a replacement:   A nasal swab is the gold standard for diagnosing COVID-19.     The  served as the gold standard for all scales from 1889 until 2019.   The best substitute I can think of is &quot;official,&quot; as in &quot;a nasal swab is the official way to diagnose COVID-19&quot;. &quot;Official&quot; implies some form of uniqueness. But &quot;official&quot; implies the existence of some authority that gives it this designation (as opposed to scientific consensus). A  asks for a &quot;perhaps humorous&quot; alternative -- the suggestion there, &quot;mother of all&quot;, is not appropriate here. (There is also .)  In response to the answers and comments doubting the premise of this question: I personally have a negative association with the phrase &quot;gold standard&quot; because I know people who advocate for returning to the gold standard or its modern equivalent, , and have discussed the topic at length. Several  US  continue to advocate for a return to the gold standard. Thus asserting that no one will interpret the phrase in terms of monetary policy is simply not convincing. Homotopy equivalence of universal cover Why is $5^{n+1}+2\cdot 3^n+1$ a multiple of $8$ for every natural number $n$? While any of the other sections under &quot;Edit Profile &amp; Settings&quot; do work normally. How to fix 'android.os.NetworkOnMainThreadException'? When I learned English at school, I was taught that I should not end a sentence with a preposition.  Is it correct to end a sentence with a preposition?     To avoid starting a sentence with a conjunction, I don't write sentences similar to "I miss you. And your kisses." Is there a similar way to avoid ending a sentence with a preposition? Title page line spacing I have a simple problem but I didn't manage to find any answer for it. In my title page, I just used  \title{Name of the title}   then  \hrule height 0,1cm \leavevmode \\[0,5cm]     {\Huge{\textbf{\@title}}}\\[0,5cm] \hrule height 0,1cm \leavevmode \\[0,5cm]   to print the title of my document. But it takes more than 3 lines on the PDF file, and I have a different spacing between the 1st and 2nd line compared to the 2nd and 3rd, which is a bit frustrating.   Anyone got an idea of why ? What are the technical differences between using a DSLR viewfinder and live view? The earlier question  asks for practical advantages of optical viewfinders. But what's the technical difference, and why does a camera behave differently in each mode?  For example, on my Nikon D5100:   Slower auto-focus — I saw this explained in the related issue, but just partially: "The other major drawback of live view is that because it requires the mirror to be locked up, it means that the auto focusing mechanism is unavailable."  option not available. No . The manual says that it's dangerous to shoot direct light, when it's on live-view. There's no range finder available.   Why are all these things different?  Live view also has some advantages, like:   automatic scene selection grid lines on the screen other smaller things, like possibly easier operation on a tripod where the viewfinder might be inconvenient to look through   Some of these things, like the tripod operation, are natural. But are there technical reasons the others can't be done with an optical finder? Are there other advantages enabled by live view for technical reasons? I've found plenty of posts about changing the visibility of a block programatically.  My problem is the opposite. Given a block's ID/delta, how do I find out if it should be displayed at the current path?  I have a field collection that allows you to append a block using the Block Reference module (I preprocess the field collection and use my own template, so the logic has been taken away from the Block Reference module). It works, but I only want it to kick in if the "Show block on specific pages" rule says that block should be displayed. Auto increment filename I have a need to collect some duplicate files and want to avoid name collisions. The problem is that this collection of files might be added to by another execution of my script before the files are cleaned out and just want to continue to increment the number.  I decided on a simple until loop to increment the number like so.  until [[ ! -f ${PWD}/DUPES/${num}-$1 ]]; do num=$((num +1)) done mv --no-clobber $1 ${PWD}/DUPES/${num}-$1   I can't imagine ever exceeding 1k files in an extreme example, so my questions are...  Is this a terribly inefficient way to achieve this? Should I be parsing the existing files to get the highest leading number to start incrementing from there, or is this ok for the extreme example I put forth?  Or is there a better way all together? Something's been bugging me about SO for a while now, that is the inaccuracy of "length of time" strings.  The best example I can give at the moment is .  If you look at the length of time he's been registered, it says 1 year.  He joined the site September 2008, which is only 3 months off two years.  It's the same with answers - you can give an answer 50 seconds before anyone else, but at certain times page views will show both as "5 minutes ago" or whatever.  That's not quite as bad, but with answers being sorted "randomly by votes" by default you could lose out on votes or being marked as the correct answer to the slower answer even if they're both the same.  There's probably a better method amongst these, IMO:   "Vulgar fraction" unicode characters     Member for 1½ years   answered 5¼ minutes ago asked 3⅓ hours ago  Different units   Member for 21 months (at least it's better than "1 year") Answered 306 seconds ago (ugly, I know) Asked 64 minutes ago (also ugly)  Combining units   Member for 1 year 9 months (fits easily on the profile screen) Answered 5 minutes 6 seconds ago (probably too long) Asked 3 hours 20 minutes ago (also long)  Rounding up   I personally prefer option 1, unicode fraction characters.  "Member for 21 months" is pretty good too because it's easy to quickly work out.  "Asked 452 seconds" ago is horrible though.  EDIT: I'm completely aware of the timestamp hover because I am forced into using it so much. It's a great feature, but it's not really in your face so not everyone knows about it. I still think there's a better "glancing" solution than having to hover over the time string. Reputation does not update after community wiki is removed from a post, and does not match the value displayed in the reputation audit they were question banned; it turned out it (in part) it was because of deleted questions.   was deleted by the Community user.  I'm aware it will delete old posts that have a negative score, but for a minute or two, I was really puzzled as to why it was unilaterally deleted without some notation left.  Can we have the Community user leave a notation in a post's history as to why the post was deleted?    Something like the following would suffice:     Post deleted automatically due to having negative votes and no activity for three months My equation is not showing big bracket and not showing in next line The essential gist of my question relates to what the real, unvarnished realities are of graduate, post-graduate, research, fellowship, and professorship attainment - when you aren't a single person with no pressing demands other than feeding yourself.   [EDIT: I am humbled by the incredibly helpful and thoughtful responses so far, all of which are +1 from me. I'll leave the question open a few more days before clicking 'accept' on any one, but you all have my gratitude and I hope others will benefit anywhere near as much as I have. ]  What I don't know is what succeeding in this path truly requires in time committment and family support.   Right now I can handle a 12-14 credit course schedule with 3.7-4.0 GPA (in the top 5-10% for college and program, dean's list, etc), work part-time, and still have time to be home and not have a burn-out/breakdown - and the academics are the easy part by far! I could do more, but I need to work more too.   What I really need to know is how sharply different is the time/dedication curve of graduate work and above. I realize each step is a filter that removes most of the entrants (most people don't get a degree, most who do don't attend/finish graduate school, most who do don't seek a PhD,  most who do don't seek/gain professorship...and on and on). But I don't know how severe the stepping is - I'm not talking "tenured professor at MIT or bust" - I realize that bar is many orders of magnitude higher.  Background Details: As an "adult, returning" undergraduate student with a (disabled) wife and kids, I'm halfway through a bachelor degree now. I've located a field of study (in a focus of Industrial Engineering) that has made me very interested in altering my life path to include seeking a Masters and maybe even a PhD/professorship (my father was a professor, so its not a purely novel concept to me). I've met and spoken with my advisor and 2 other professors with experience in related fields, and all is encouraging; I have a graduate school in mind, I've received lots of great advice on preparing for school (on this forum and in person), I test well (no GRE fears remain), good grades/class rank, good prospects for future strong referal letters, etc - a comparitively charmed existence for which I am very grateful! I just have to investigate whether I can make it through with an intact, cared for family!  Any guidance you could provide me, and other aspiring "non-traditional" students alike, would be appreciated immensely - thank you! I'm buying a bus ticket to get to Boston 23 hours before my flight to ensure I reach my departure airport on time.  If I didn't arrange bus transport well in advance, what legal remedies would I have if a delayed bus caused me to miss my flight? 4 fast cores or 6 slower ones? After installing the latest Ubuntu I find that the icons in the applications folder (viz. /usr/share/applications), when opened in the Files application (Nautilus), come up as a blank page with a gear in the middle. If I try to open them I get a read only text box. I was just testing Eevee and found that you cannot view a glass object through another glass. Is there any workaround? It works fine only If I rotate the glass window or put my camera to the back side of the scene How to minimize the ink-to-data ratio for pgfplots