Find the interval of convergence for $\sum_{k=0}^{\infty}\frac{x^k}{k!}$.  I tried using the ratio test to find the radius of convergence.  \begin{align} \lim_{k\to\infty}\dfrac{\left|\frac{x^{k+1}}{(k+1)!}\right|}{\left|\frac{x^k}{k!}\right|} &amp;= \lim_{k\to\infty}\left|\dfrac{x}{k+1}\right| =0 \end{align} Doesn't this tell us that we can choose any $x\in\mathbb{R}$ because $\lim_{k\to\infty}\left|\dfrac{x}{k+1}\right| =0$ for any $x$?
Suppose one is trying to find the radius of convergence of  $$ \exp(x)=\sum\limits_{k=0}^{\infty} \frac{x^k}{k!} $$  This "proof" was given in the lecture: $$ \begin{align} r =&amp; \frac{1}{\limsup\limits_{n \rightarrow \infty}\sqrt [n]{|a_n|}} \\ =&amp; \frac{1}{\limsup\limits_{n \rightarrow \infty}\sqrt [n]{\frac{1}{n!}}} \\ =&amp;\lim\limits_{n \rightarrow \infty} \sqrt [n]{n!}\\ r=&amp; \infty \end{align}$$  Other proofs involving the ratio test also seem to inspect the convergence of $\frac{1}{k!}$ instead of $\frac{x^k}{k!}$. Why is one allowed to substitute $\frac{1}{k!}$?
Biber 2.11: Data file ... cannot be read in encoding UTF-8 ... ill-formed UTF-8 octet sequence ... at ... Slurper.pm line 63 I have just changed computers, and running Biber 2.11 now produces this error:  Utils.pm:209&gt; ERROR - Data file ... cannot be read in encoding 'UTF-8':  Can't decode ill-formed UTF-8 octet sequence &lt;96&gt; at C:\ ... Slurper.pm line 63   My .bib file is huge, having accumulated over years.  Thus, I suspect that there's something malformed in it.  Is there any rapid way to trouble-shoot it?  The new machine runs Win 10, Biber 2.11, XeLaTeX 2018.10.31, LaTeX2e 2018-04-01 and TeXLive 2018.  The old machine ran Win 7 and slightly earlier versions of all of the above.  A MWE of the driver .tex file:  \documentclass{article} % also with beamer  \usepackage[utf8]{inputenc} % problem arises whether or not include this \usepackage[backend=biber,bibencoding=utf8]{biblatex} %  problem arises whether or not include this \addbibresource{bib-mwe.bib}  \begin{document}  cite reference \cite{mal-rwws}  \printbibliography  \end{document}    Thanks! If $\gcd(a,b)=1$, $\gcd(a,y)=1$ and $\gcd(b,x)=1$ then prove that $ax+by$ is prime to $ab$ If $\gcd(a,b)=1$, $\gcd(a,y)=1$ and $\gcd(b,x)=1$ then prove that $ax+by$ is prime to $ab$.  I tried assuming Diophantine Equations for all the relations and representing everything in terms of $a$, but that didn't lead me anywhere. There must be an intuitive and elegant approach right? Is the sign of principal components meaningless? Does the sign of the vectors returned by PCA have any significance after I center my variables? NFL overtime: after a field goal, clock runs out on second team's possession Under the NFL's current "modified sudden death" overtime rules, if the first team scores a field goal, the second team is entitled to one more possession.  What happens if the clock runs out during that possession?  Suppose the score is tied 20-20 at the end of the 4th quarter. Team A wins the overtime coin toss and receives the kickoff to begin overtime.  After a long slow drive, Team A scores a field goal to make the score 23-20.  Now they kick off to Team B.  Team B is not able to score, and the overtime period ends with Team B still in possession.  What happens?   is not very explicit on this point.  Article 4 (a) says:     There shall be a maximum of one 15-minute period, even if the second team has not had an opportunity to possess the ball or if its initial possession has not ended. If the score is tied at the end of the period, the game shall result in a tie.   I think this means that Team A wins, 23-20, but I am not completely sure.  Has this ever happened in a regular-season NFL game? text processing rows to columns for a block There have been a number of cases where I would like to thank someone for providing me with a thorough response and give them feedback beyond the vote up. Often I wish people would tell me why they voted me up. Not that the thank you part is necessary but it sets it apart from the other comments related to answering the question itself.  In this type of situation is a "Thank you" acceptable in a comment?     For more information, see  in the . Why is it disadvantageous in graduate admissions to have a higher degree? indicates that holders of a master’s degree are held to higher standards when it comes to graduate admissions. The stated requirements for some graduate programs () also outright exclude applicants with doctoral degrees. These seem to be saying that holding a higher degree prior to application is a disadvantage (a fatal one in the case of a PhD). This also applies to private universities: Stanford  (at least in computer science); .  I don't understand why:   If there are two applicants, Alice and Bob, who have achieved largely similar things, it seems logical that both should have a roughly equal chance of being admitted. However, based on JeffE's answer, if Alice has a Masters degree while Bob does not, it's Bob that gets admitted instead of Alice. JeffE's answer even indicates that if Alice has achieved slightly more than Bob, it's still Bob that gets admitted.  If graduate admissions is looking for students that are capable of succeeding in its program, then Alice should be more likely to succeed than Bob since she's already had some experience of graduate study. This should be even more the case if Alice already has a PhD – if she can do it once, she should be able to do it again.    JeffE's answer indicates that since Alice has had more opportunities than Bob, she should also be held to higher standard. However it should also be true that Alice has already achieved what Bob has yet to and in fact might not be able to achieve. Further, JeffE's answer seems to be saying that Alice should do the academic equivalent of financial engineering and pass her graduate-level courses &amp; thesis without getting a master’s degree even if she qualifies for it, which sounds silly.  As for applicants with PhD degrees, my first guess would be that this is because , and the university is acting in the applicant's best interests. However if this is the case, then this is effectively telling the applicant "I know what you want better than you do", which also sounds silly. UC Berkeley's requirements also seem to be saying that if someone with a CV worthy of a tenured position were to apply for a PhD student position, the university will say "no" even though it's clearly better in their best interests (in terms of research output) to say "yes".   Presumably I'm misunderstanding something because if this behavior were irrational, it would already have been changed. What am I missing?  EDIT: I wonder if things are different in programs where the student is involved in research directly without spending time on coursework? Is there any matrix $2\times 2$ such that $A\neq I$ but $ A^3=I$? Is there any $2 \times 2$ matrix $A \neq I$ such that $A^3=I$?  In my opinion: no. Thank you very much. I run windows 7. I run a Minecraft server, but I can only host around 9 players before lag appears.   I heard that Ubuntu can run a server with much better performance than Windows 7. I kind of like the whole idea of Ubuntu, but I don't want to change my OS to Unix or anything else.  Basically, my question is, does running a server (specifically and Minecraft server) run easier and faster on a Ubuntu server? On top of that, how would one set that up? Extract URLs from text in PHP How to manage memory alignments and generic pointer arithmetics in a portable way in C? Beamer text and image on the same slide Why are the solutions of polynomial equations so unconstrained over the quaternions? Calculating the average area of the states listed in states.txt Need someone to calculate the average area, in square miles, of the states listed in states.txt. States.txt (file): State           Population   SqMi      Region  ------------------------------------------ California           39.37   163.7       West Texas                29.36   268.6      South Florida              21.73    65.7      South New_York             19.33    54.7  NorthEast Pennsylvania         12.78    46.1  NorthEast Illinois             12.59    57.9    Midwest Ohio                 11.69    44.8    Midwest Georgia              10.71    59.4      South North_Carolina       10.60    53.8      South Michigan             9.97    96.7    Midwest  Output should look like this: Average Area: 82.85 square miles  Note my awk program will be ran like this: awk –f &lt;file.awk&gt; states.txt What is vertex and edge shading? What's the English equivalent of the Japanese saying, “A fart ruins 100 days of sermons by the priest (bishop)”? I was amused by the expression "Paid a penny and only farted" (related by @FumbleFingers), which suggested a similar Japanese saying: 大山鳴動鼠一匹 - "Find only a small mouse coming out after hearing rumblings and experiencing shaking of the big mountain."  It’s humorous. I chuckled when I read it, and I wrote to the author that I love it.  With that said, the word "fart" reminds me of another Japanese proverb, 百日の説法屁一つ- "A fart ruins 100 days of sermons by the priest", which means that a small mistake (the sound of a fart in front of the congregation) brings all his efforts to naught.  It can’t happen in big Western churches. But it could well happen in a small wooden temple where a priest would preach to 30 – 50 provincial followers. Can you imagine how the proud priest being called 'the venerable' would be embarrased when his loud fart is heard by all his followers during his serious sermon and they start to giggle here and there in the hall?  We also have a similar saying" 九仞の功を一簣に欠く, which literally means "Collapse a 9- meter sand tower with the last pile of sand."  What is the English equivalent of the saying "A fart ruins 100 days of sermons by the priest (bishop): a small mistake depreciates the value of (vitiates) all labor and effort to naught," preferably in such a humorous way? Functions Which are non differentiable on a Given Set. Not every open set in $\mathbb{R}^n$, for $n\geq2$, is a disjoint union of countable open balls. I know that every open set in $\mathbb{R}$ can be decomposed as the union of countably many disjoint open intervals and I'm trying to show that this property no longer holds in higher dimensions. For this I'm trying to show that there is no countable, pairwise disjoint open balls whose union is the open square $(0, 1)^2 ⊂ \mathbb{R}^2$. I know that it is because of the corners, but I don't know how to write this in mathematical terms. Thanks for any help. Access 2007 Work in a separate file using ACCDE Have been working in an Access 2007 application for over 3 weeks and it should be ready now. Anyway, I know (and tried) that "Make ACCDE" can actually split the raw data from the application itself so you can run it using the Access RunTime.  The question is... can I deploy my app for general use and modify a separate copy of the forms/views inside the application to deploy it when it´s ready without touching the data? How to prevent the paragraphs to spread out over the page? I have a problem. I have several large equations within the text. And when LaTeX does not manage to put the equation on the same page and moves it to the next one, he spreads the paragraphs of the text in the first page. How could I prevent this? I know about \pagebreak and \newpage, but I do not want to force the \newpage, as the text may move (if I add something later or so on) and the pagebreaks may not be valid. I would like to make LaTeX not to spread the paragraphs over a page automatically. Is it possible?  I tried to make an example:  \documentclass[a4paper,12pt,twoside]{book} \usepackage{amsmath} \begin{document}  LaLaLa\\\  LaLaLa\\\  LaLaLa\\\  LaLaLa\\\  \begin{eqnarray} 123&amp;+&amp;123 \\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ \end{eqnarray}  LaLaLa\\\  LaLaLa\\\  \begin{eqnarray} 123&amp;+&amp;123 \\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ \end{eqnarray}  \end{document}