Is it possible to express $0.999...$, a repeating number, as a fraction? Or as a ratio of two numbers?  Basically all (my) attempts at the problem cancels all the terms and returns $1$. Is it even possible? Or has it been proven to be an exercise of futility?
I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is?
I have seen similar posts but none seem to work. I simply need to remove the dots in the ToC between the chapter titles and the page numbers. I think the page numbers should line up (as if there were dots). This is using the memoir class. Here is my ToC info:  %%%% Do the ToC \let\oldtoc\tableofcontents \renewcommand{\tableofcontents}{\clearpage\pagestyle{toc}\oldtoc}  \renewcommand{\contentsname}{TABLE OF CONTENTS} \renewcommand{\listfigurename}{LIST OF FIGURES} \renewcommand{\listtablename}{LIST OF TABLES} \renewcommand*{\tocheadstart}{\vspace*{-\topfiddle}} \renewcommand*{\aftertoctitle}{\thispagestyle{plain}%   \par\nobreak \mbox{}\hfill{\normalfont Page}\par\nobreak} \renewcommand*{\cftchapterfont}{\normalfont} \renewcommand*{\cftchapterpagefont}{\normalfont} \renewcommand*{\cftchapterleader}{%   \cftchapterfont\cftdotfill{\cftchapterdotsep}} \renewcommand*{\cftchapterdotsep}{\cftdotsep} %\renewcommand*{\cftchaptername}{CHAPTER~} %%% no extra space before the entry \setlength{\cftbeforechapterskip}{0pt plus 0pt} %% no extra 'chapter' space in LoF/LoT \renewcommand*{\insertchapterspace}{}   I tried everything that was suggested. Nothing seemed to make any difference. I am stuck with this format. Any suggestions would be greately appreciated! Query Activity for ORDER BY xxx DESC LIMIT I'm trying to create a query activity in order to show, from a Data Extension, only the last 5 articles ids. I tried with the formula ORDER BY xxx DESC LIMIT 0,5 as in SQL but it is not working and it is returning this exception:     The ORDER BY clause is invalid in views, inline functions, derived tables, subqueries, and common table expressions, unless TOP or FOR XML is also specified   How can I use ORDER BY in a Data Extension? Avoiding Windows 10 Creator, but not other updates I do as follows  $(1+x)^n= \binom{n}{0}+\binom{n}{1}x+\binom{n}{2}x^2+...+\binom{n}{n}x^n$   differentiating in both sides  $n(1+x)^{n-1}= \binom{n}{1}+2\binom{n}{2}x+3\binom{n}{3}x^2...+n\binom{n}{n}x^{n-1}$  then, for $x=1$  $n \cdot2^{n-1} = \sum_{k=1}^{n}k\binom{n}{k}$  From here, I don't know how to proceed, I have tried to multiply both sides for $(1+x)$ to make $n \cdot2^{n}$ but then the summatory changes, I don't know how to prove the equality. Coding for an ordered covariate I am performing ordinal regression, I have 5 response categories and several predictors both continuous and categorical. I would like to add a predictor which is categorical but ordered (1, 2, 3, 4). I don't think it would be appropriate to apply the usual dummy coding for unordered categorical predictors, but when I searched for how to code this I did not find much information.  In  "linear coding" or "assuming linearity of the predictor effect" is mentioned, but without further details.  Does it mean I just use my ordered values as they are, i.e. use them as a continuous variable? I have asked this question on SO but perhaps that was the wrong location. Apologies for double posting but the question may have a better audience here.  The original question is:   And the text is below. Thanks.    When running the postgresql [9.3] "COPY TO" functionality, the postgres user creates a file. I have told it to put the file in the tmp directory (although this can be changed if needed).  This is run as part of a cron job, the user is an "export" user set up especially for this task. The end goal is for the exported file to end up in /home/export/created_file.csv.  1) postgres can't stash the exported file straight into the home directory of another user without setting up the folder to be overly permissive.  2) the export user can copy the file from /tmp to it's home directory, but then can not clean up (delete) the postgres owned file in /tmp.  This is a daily process and so old files will need to be deleted in some way. I can add a cron job to clean up the files afterwards but I was hoping for a cleaner solution - either a way to change the permissions on the file that postgres creates, or some way to give the export user the additional permission of being able to delete that file in advance of it's creation.  Giving the export user the same permissions as postgres in general however isn't an option, as it would then have free reign over the database.  Any advice, be it unix based or postgres based would be very welcome, thanks. In a 'bulleted' list should items that happen to be complete sentences end with a period? Using global variables in a function How can I create or use a global variable in a function?  If I create a global variable in one function, how can I use that global variable in another function? Do I need to store the global variable in a local variable of the function which needs its access? Need help in proving that $\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$ We need to prove that $$\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$$  I have tried and it gets confusing. How can I obtain the limit of the series     $$\sum_{n=1}^{\infty} \frac{2^{[\sqrt{n}]}+2^{-[\sqrt{n}]}}{2^n}$$   Where $[\ \ ]$ is . Is it possible for some integer $n&gt;1$ that $2^n-1\mid 3^n-1$ ?   I have tried many things, but nothing worked. The Order of Modification in English Nouns, Preceding or Succeeding? As I don't know the exact linguistic terms, what I mean my "preceding" and "succeeding" in modifying nouns is as follows.  Preceding : delicious food, long way, kind person, et cetera Succeeding : faculty of science, land of despair, a book that changed my mind, et cetera  English seems to accept both ways of modification.  Sometimes, either way will not make a difference. department of science == science department a burning book == a book burning  At other times, changing the order will still make sense but be somewhat awkward or bizarre. Kentucky Fried Chicken ?= Chicken Fried in Kentucky free as in free speech ?= as-in-free-speech free  When writing English, I tend to use more the preceding modification style because my native language Korean is quite strictly precedingly modifying. "빠른 판단을 바탕으로 한 재치있는 응수" directly translates to "a quick-decision-based clever response" rather than "a clever response based on a quick decision"  French as well as other Latin-originated languages seems to be succeedingly modifying. Académie Française, Poulet Frit du Kentucky  while German and Dutch more often use preceding modification.  Well, this mixture in English allows rich style with various ways of expression, but it does often make a non-native English speaker be a little confused. Is there any subtle but general difference between these 2 ways? Or is it just simply case-by-case. Any guidance will help. I need to get MODIS NDVI (MOD13A3), but output from EarthExplorer or Daac2Disk HDF format is a numerical range is -2000 to 9996.   But I need for futher work basic range -1 to 1.   Is there any conversion method? I was trying to solve the following:  Given: $0 &lt; a &lt; b$ and $n&gt;1$, prove: $$na^{n-1}(b-a) &lt; b^n-a^n &lt; nb^{n-1}(b-a)$$  I managed to get this far using the mean value theorem: $$a^n(b-a)&lt;b^n -a^n&lt;b^n (b-a)$$ Any idea how to continue? What is the formula for armor/magic resist reduction? As far as I can tell, it's something like this:  damage reduction = armor / (armor + 100)   Is this accurate? What (if any) countries are either open to or have set definitive plans to allow international tourism? When in many rooms, user's info goes off screen So, I wanted to check what low-number rooms are still active so I joined a bunch of rooms in the chat.  This is the result:    Bug also exists when expending rooms on the side:    Since the position is fixed I can't scroll down.  Proposed fix:  Only show the first 10 rooms, and show ... pointing to the user's profile for other rooms. Create an interface which is similar to the material list box Saving Working Files as PSD or TIFF? I was reading a few articles about how people saved their working files and it got me thinking about my own method. I used to save everything as a PSD, but recently I have started saving my working files as TIFF's. I use Windows, so I don't know if it's different for Apple users but with a PSD, all I can see is that "PSD" icon. But a TIFF file shows up as a normal image (that I can edit), so I found it really great when I have a lot of working files for one project and could just quickly preview it without opening the program.  That being said, I am no expert in which method is better. And I know a lot of people here are pretty seasoned users in all things Adobe, so I was wondering what your two cent on the whole topic is?   Is there really any difference to saving it as a PSD or TIFF? Is there something I should specifically know about saving my files as a TIFF rather than a PSD? Xubuntu 18.04 Persistent Live USB: Will persistence be carried upon full installation to a machine? I've created an Xubuntu 18.04 persistent live USB stick and installed several softwares for web development, such as XAMPP, Sublime Text 3, Google Chrome, etc.. I did some configurations as well in the ~/.bashrc file.  I am wondering if these installation and configuration in my persistent live USB stick be carried over in a full installation in to a new machine using it.