Let $S = [{(-1)^n+ \frac{1}{n}}$] and $n \in \mathbb{N}$   Find the accumulation point of S.  I don't think there are any, but I am not sure. Can anyone shed some light on this?
I was recently asked to find the accumulation points of the set $$S = \{(-1)^n + \frac{1}{n} \mid n \in \mathbb{N}\}$$  I answered that the accumulation points are $\{-1,1\}$, because despite the fact that $\frac{1}{n}$ diverges, we can still use $\frac{1}{n}$ to find an element arbitrarily close to either $1$ or $-1$, i.e. to show that there is an element in an arbitrarily small neighborhood of $1$ or $-1$. (No proof was required for this question --- this is my intuition.)  Am I right at all?
The question Is there any way to automatically scale and put an entire document on an A0 page? The thesis for which this is to be done is available as LaTeX code, as well as PDF output. Of course I know it is possible to do this manually, but I would prefer a LaTeX solution.  A small background Our university introduced a new page limit for master theses. This is not exactly appreciated by the teaching staff as well as students. To make a little joke, we decided to try to arrange an entire thesis (of approximately 200 pages) on a A0 poster, put it up in the hallway, and put a magnification glass next to it with a note saying: &quot;New page limit introduced: 1 page maximum for all upcoming theses!&quot;  The final result As some of you asked for this: below you can see a picture of the final result. (I &quot;professionally removed&quot; some of the sensitiv information.) We presented the poster during my final presentation and everyone liked it a lot! Now, it is on display next to my office and is still attracting some attention once in a while. I find in many places mentions of pre-amplifiers being used to boost audio signals prior to feeding to a power amplifier.  Wikipedia's article on  gives some hints at their usage, but it doesn't really explain why it would be necessary to have two amplifiers in series rather than a single one with greater gain.  If a pre-amp is just a power amp with a smaller gain, what's the point, and why the distinction between the two kinds? Is the distinction in function, or simply purpose? How do I do Combinatorics / Counting? If someone could give answers and explain, it would be greatly appreciated. Help required studying for a final.  One hundred tickets, numbered 1,2,3,…,100, are sold to 100 different people for a drawing. Four different prizes are awarded, including a grand prize (a trip to Tahiti).   A) How many ways are there to award the prizes?  B) How many ways are there to award the prizes if the person holding ticket 47 wins the grand prize?  C) How many ways are there to award the prizes if the person holding ticket 47 wins one of the prizes?  D) How many ways are there to award the prizes if the person holding ticket 47 does not win a prize?  E) How many ways are there to award the prizes if the people holding tickets 19 and 47 both win prizes?  F) How many ways are there to award the prizes if the people holding tickets 19, 47, 73, and 97 all win prizes?  G) How many ways are there to award the prizes if none of the people holding tickets 19, 47, 73, and 97 wins a prize?  H) How many ways are there to award the prizes if the grand prize winner is a person holding ticket 18, 47, 73, or 97?  I) How many ways are there to award the prizes if the people holding tickets 19 and 47 win prizes, but the people holding tickets 73 and 97 do not win prizes? Alternate moves, first one to roll a 6 wins: what's the probability of winning? Question: Players A and B play a sequence of independent games. Player A starts and throws a fair die and wins on a "six". If she fails then player B throws the same die and wins on a "six" or fails otherwise. If she fails then player A gets the die and throws again, and so on. The first player to throw a "six" wins. Find the probability that player A wins.  Here is what I have done:  A wins right away: $\dfrac{1}{6}$  A fails B wins: $\dfrac{5}{6}\times \dfrac{1}{6}$  A fails B fails A wins: $\dfrac{5}{6}\times \dfrac{5}{6}\times \dfrac{1}{6}$  Unlike another similar question which the player B can win on "five" and "six" after player A fails on the first throw, in this case, the games do not seem to have an endpoint. How should I calculate the probability that player A wins then? Grep in couple thousands files I have directory with cca 26 000 files and I need to grep in all these files. Problem is, that I need it as fast as possible, so it's not ideal to make script where grep will take name of one file from find command and write matches to file. Before "arguments list too long" issue it took cca 2 minutes to grep in all this files. Any ideas how to do it? edit: there is a script that is making new files all the time, so it's not possible to put all files to different dirs. In addition to whatever files I may choose to exclude, what files does the system exclude by rule or classification?    It might make sense not to backup Trash or Caches, but I'm looking for a way to tell what files are skipped "behind the scenes." Knowing if the system maintains an exclusion list or exclusion pattern would be very helpful in managing larger groups of macs. Do I have to buy or need a 3D-enabled phone to use Google Cardboard? Google cardboard comes with 3D glasses. Does that mean that we will be able to watch 3D videos using a smartphone like the Moto-G (1st edition) which is not 3D enabled? Why does LuaTeX plus breqn quadruple output of \text{} when in display-math mode? MWE needs to be compiled with LuaTeX to reproduce the problem:  \documentclass[11pt]{report}% \usepackage{amsmath,mathtools} \usepackage{breqn}                          \begin{document} $J^{\ast} = \text{JFK} \to \text{NO} \to \text{SFO}$ \[ J^{\ast} = \text{JFK} \to \text{NO} \to \text{SFO} \] \end{document}   Gives (no compile error or any warning) the following strange output    I have to use breqn package. If I remove it, then I get the expected output    Is there a way to work around this, so I can use breqn and get correct output?  TL 2015. filelist: (compiled with lualatex)  This is LuaTeX, Version beta-0.80.0.1 (TeX Live 2015) (rev 5863)   *File List*   report.cls    2014/09/29 v1.4h Standard LaTeX document class   size11.clo    2014/09/29 v1.4h Standard LaTeX file (size option)  amsmath.sty    2013/01/14 v2.14 AMS math features  amstext.sty    2000/06/29 v2.01   amsgen.sty    1999/11/30 v2.0   amsbsy.sty    1999/11/29 v1.2d   amsopn.sty    1999/12/14 v2.01 operator names mathtools.sty    2015/11/12 v1.18 mathematical typesetting tools   keyval.sty    2014/10/28 v1.15 key=value parser (DPC)     calc.sty    2014/10/28 v4.3 Infix arithmetic (KKT,FJ)  mhsetup.sty    2010/01/21 v1.2a programming setup (MH)    breqn.sty    2015/08/11 v0.98d Breaking equations    expl3.sty    2015/12/20 v6326 L3 programming layer (loader)  expl3-code.tex    2015/12/20 v6326 L3 programming layer  l3pdfmode.def    2015/11/11 v6250 L3 Experimental driver: PDF mode flexisym.sty    2015/08/11 v0.98d Make math characters macros   cmbase.sym    2007/12/19 v0.92 mathstyle.sty    2015/08/11 v0.98d Tracking mathstyle implicitly graphicx.sty    2014/10/28 v1.0g Enhanced LaTeX Graphics (DPC,SPQR) graphics.sty    2014/10/28 v1.0p Standard LaTeX Graphics (DPC,SPQR)     trig.sty    1999/03/16 v1.09 sin cos tan (DPC) graphics.cfg    2010/04/23 v1.9 graphics configuration of TeX Live   pdftex.def    2011/05/27 v0.06d Graphics/color for pdfTeX infwarerr.sty    2010/04/08 v1.3 Providing info/warning/error messages (HO)  ltxcmds.sty    2011/11/09 v1.22 LaTeX kernel commands for general use (HO) supp-pdf.mkii pdftexcmds.sty    2011/11/29 v0.20 Utility functions of pdfTeX for LuaTeX (HO) ifluatex.sty    2010/03/01 v1.3 Provides the ifluatex switch (HO)    ifpdf.sty    2011/01/30 v2.3 Provides the ifpdf switch (HO) luatex-loader.sty    2010/03/09 v0.4 Lua module loader (HO) epstopdf-base.sty    2010/02/09 v2.5 Base part for package epstopdf   grfext.sty    2010/08/19 v1.1 Manage graphics extensions (HO) kvdefinekeys.sty    2011/04/07 v1.3 Define keys (HO) kvoptions.sty    2011/06/30 v3.11 Key value format for package options (HO) kvsetkeys.sty    2012/04/25 v1.16 Key value parser (HO) etexcmds.sty    2011/02/16 v1.5 Avoid name clashes with e-TeX commands (HO) epstopdf-sys.cfg    2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live  ***********   Update  I found another case of duplication when using breqn, which does not involve \text at all. But the solution provided below also fixes this issue. I am adding this second case here for reference just in case it might be useful. Here is MWE  \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage{amsmath,mathtools} \usepackage{breqn} \begin{document} \[ \underset{\mathbf{S}} {\operatorname{arg\,min}}      \sum_{i=1}^{k} \sum_{\mathbf x \in S_i}       \left\| \mathbf x - \boldsymbol\mu_i \right\|^2  \] \end{document}   The output is    After adding the patch in the answer below the correct output is now given, which is Groups of prime squared order Is there a DSLR fixed lens camera? We were asked to prove the following:  Let $ A $ be an $n \times n$ matrix with $n$ distinct real eigenvalues. If $AB=BA$, show that $B$ is diagonalizable.   It was suggested I show that an eigenvector of $A$ is also an eigenvector of $B$. I am both having trouble doing this and failing to see how I would complete the proof after. Any help would be appreciated.  Thanks This was a question from my textbook. This is how I approached it.   For the first part, you can do c(10,5) so you choose 5 of the 10 to be on a team.   For the second part, you have 10 ways to choose the "best player" and you multiply it by c(10,4) to pick the other 4 players. However, I am not sure how you can exclude the "worst player". Although this sounds subjective, I believe the question is saying that the "best" and "worst" players are distinct, whereas the other players aren't. Why is the space between words in my source code listing so big? I used the following code to list my java code, but the it looks like there's too much space between words. What can I do?  % source code listing \usepackage{listings} \usepackage{color} \usepackage{xcolor}  \usepackage{caption}  % citation style \usepackage{apacite} \DeclareCaptionFont{white}{\color{white}} \DeclareCaptionFormat{listing}{\colorbox{gray}{\parbox{\textwidth}{#1#2#3}}}  \begin{document} \begin{lstlisting}[label=some-code,caption=bind to service call] bindService(new Intent(this, Recognizer.class), recConnection, Context.BIND_AUTO_CREATE); \end{lstlisting} \end{document}   The snapshot of what I got. The code not completely shown, and space too wide, in function bindService. Is $n^{th}$ root of $2$ an irrational number? Possible Duplicate:          Will every $n^{th}$ root of $2$ be an irrational number? If yes, how can I prove that? .Net 'Any Framework' configuration My PDO Statement doesn't work How to partition an encrypted disk Suppose $\{a_n\}$ is a sequence such that the subsequences $\{a_{2n−1}\}$ and $\{a_{2n}\}$ converge to the same limit, say $a$. Show that $\{a_n\}$ also converges to $a$. Why not use the Lagrangian, instead of the Hamiltonian, in nonrelativistic QM? Pattern matching with LIKE, SIMILAR TO or regular expressions in PostgreSQL I had to write a simple query where I go looking for people's name that start with a B or a D :   SELECT s.name  FROM spelers s  WHERE s.name LIKE 'B%' OR s.name LIKE 'D%' ORDER BY 1   I was wondering if there is a way to rewrite this to become more performant. So I can avoid or and / or like?