Prove $ \sum\nolimits_{n=1}^{ \infty } u_n(x)$ converges uniformly at $[0,1]$, when $u_n(x) = (-1)^n*(1-x)*x^n$ I can't understand how to prove that $ \sum\nolimits_{n=1}^{ \infty } u_n(x)$ converges uniformly at $[0,1]$, when $u_n(x) = (-1)^n*(1-x)*x^n$. I tried using Weierstrass M-test - bounding those functions with a converging series $ \sum M_n$, which will result in $ |u_n(x)| \leq \sum M_n$. But after 1 and 1/2 hours I couldn't find the right usage here. I'm aware of the other question linked to that, but it doesn't answer exactly the same thing. Would much appreciate help. Thank you!
Uniform convergence of $\sum (-1)^nf_n(x)$ on $[0,1]$ where $f_n(x)=x^n(1-x)$. Let $f_n(x) = x^n(1-x)$ and $\sum (-1)^nf_n(x)$.  I showed that this series point wise.  Case1) $x=1$ $$ f_n(1)=0 \, , \sum{(-1)^nf_n(1)} = 0 $$ Case2) $x\neq 1$ $$|f_n(x)| \leq |x|^n$$ since $$\sum |x|^n$$ converges, By abosultely convergese test $\sum (-1)f_n(x)$ converges.  I want to show that this series uniformly converges on $[0,1]$.
In PostgreSQL, what is the difference between a "Stored Procedure" and other types of functions? I understand the distinction between:   Scalar functions Set-Returning-Functions (SRF)s Internal functions Window functions Aggregate functions of all sorts User-Implemented functions (which in PostgreSQL can be implement in any language) Etc.   In SQL Server "Stored Procedures" are permitted through EXEC. What does that provide over any other function executed with SELECT that returns NULL?  When PostgreSQL gets Stored Procedures what will they bring me, and what is the formal distinction if any between a function and a stored procedure in the spec?  I read To ensure that a new chapter starts on a right-hand page, the book class (and the report class with the options twoside and openright) will insert blank pages between chapters if necessary. However, these pages still contain the headers and footers as defined by the class. How can I remove those?  Note: I know that I can manually add \clearpage\thispagestyle{empty} at the appropriate points of my document. However, this seems awkward and is prone to fail if I change my document later. I'm looking for automatic solutions. How to search entire hard drive for a file? Where are the outliner toggles in 2.8? The selection toggle arrows in the Outliner disappeared. Is there some preference in the settings that has to be turned on? Variable Sized Arrays vs calloc in C On the discussion of dynamic memory here:   The author states:     A memory block like this can effectively be used as a more flexible array. This approach is actually much more common in real-world C programs. It's also more predictable and flexible than a "variable size array"   The type of memory block he is talking about is as such:  const int size = 5; int * array = calloc(size, sizeof(int));   and then using another pointer to iterate over the array:  int * index = array; for (i = 0; i &lt; size; i++) {     *index = 1; // or whatever value     index++; }   My question is how is this method better than a standard variable sized array like this?:  int array[variable];   or dynamic:  char name[] = "Nick";   The author doesn't really shed much light as to why I should prefer the former method to the latter. Or more specifically: How is it more "predictable and flexible"? Should we use 'Title Case' or 'Sentence case' for headlines and buttons? I am wondering if we should use  or  for buttons and headings in websites and web/mobile apps? Title Case: Manage my Account Delete my Account  Sentence case: Manage my account Delete my account  In  I found the recommendation 'choose your style and be consistent'. This is it? Aren't there any other (additional) recommendations or sources. Like for specific use cases or apps where the one style would be preferred over the other? Edit: The question refers primarily to English, but differences in other languages are also good to know. Physically what does warping (of space-time) mean? So there's general relativity and Einstein's field equations that tell us "mass(or equivalently energy) warps space-time, and the warping tells mass how to move", but I'm still having trouble understanding how space (a thing I conceive to be 'empty' or nothingness) can be 'warped'.  After researching books, on-line articles, and this stack exchange itself I can't seem to find any descriptions or discussions that don't immediately resort to jumping into GR and its mathematics to provide a more physical intuitive understanding of what it means to for space to be warped.  Is it space, this empty thing, that's getting warped. Or rather is it some field (Higgs field?) that exists around matter that's getting warped?  Although this question appears similar to  , it's asking if it's empty space that's being warped by matter or rather a field in space that's being warped (the Higgs Field?). The other question and all 10 answers addressing it do not address fields vs. empty space - What's really getting warped? Removing Firefox in Ubuntu with all add-ons like it never existed I have been using Firefox for a long time in Ubuntu, I downloaded some add-ons but it made firefox extremely slow, even if I remove them nothing will change.  I tried to remove Firefox also, but re-installing it will somehow recover the same Firefox I had- it seems it write some info somewhere that even if you remove Firefox, it will keep them.  Moreover, I am not able to use see any flash videos or website with Firefox(chrome works fine).. Honestly I tampered with plug-in settings in Firefox that this happened. But right now I can't do anything to overwrite the setting...  My question is how I can remove Firefox in a way to remove any previous traces and re-install a new version like for the first time? How to insert code with accents with listings? I'm trying to insert source code in LaTeX with using listings, but my code has accents and pdflatex gives me this error:  Unicode char \u8:�\expandafter not set up for use with LaTeX.   This is a code the reproduces the error.  example.tex  \documentclass{article} \usepackage{listings} \usepackage[utf8]{inputenc} \usepackage[spanish]{babel}  \begin{document}  \lstinputlisting[language=C++]{code.cpp}  \end{document}   code.cpp  // Código con una tilde   What can I do to generate the PDF correctly? New Beamer Version: Transparent Title Page Boxes Removing OS from Dual Boot computer, and recovering HD space. Is there any way to block websites without Root access I am searching for a way to block websites without having root access. Is there any way to do it? Any help will be appreciated. Listings - Only highlight between delimiters I want to implement a function name color highlighting and I use the listings package. My 'regex' for this is def %s( where def (including space) is the first delimiter %s is the function name and ( is the second delimtier. Now my aim is to color only %s in purple. So basically I want to only touch the part between the delimiters and the delimiters should stay untouched.  Here is what I tried so far:  \documentclass{article} \usepackage{xcolor,listings} \usepackage[ngerman]{babel} \usepackage{fontspec} \setmonofont{Courier New}  \begin{document}  \lstset{language=Python,     escapebegin=\color{orange},     keywordstyle=\color{blue}\bfseries,     commentstyle=\color{orange!80!red},     stringstyle=\ttfamily\color{gray},     %moredelim = [is][\color{blue}\bfseries def \mdseries\color{purple}]{def\ }{(},     moredelim = [s][\color{purple}]{def\ }{(},     showstringspaces=false}‎ \lstset{literate=%    *{0}{{{\color{red}0}}}1     {1}{{{\color{red}1}}}1     {2}{{{\color{red}2}}}1     {3}{{{\color{red}3}}}1     {4}{{{\color{red}4}}}1     {5}{{{\color{red}5}}}1     {6}{{{\color{red}6}}}1     {7}{{{\color{red}7}}}1     {8}{{{\color{red}8}}}1     {9}{{{\color{red}9}}}1 }  ‎‎\begin{lstlisting}[basicstyle=\ttfamily,mathescape] def fancy_function_name(page_url = 'tex.stackexchange.com'):     return page_url.split('.')[0] \end{lstlisting}  \end{document}   This code will generate:   You can see that it matches the right place but is too greedy and colors the delimiters too. So I thought about the uncommented line: %moredelim = [is][\color{blue}\bfseries def \mdseries\color{purple}]{def\ }{(}, If I use this one and uncomment the line below I get this result:    It's nearly what I wan't but I cannot add the last, now missing ( in black color.  Does anyone has a solution for this without pretouching the code and add all the @@ symbols? In a topological space, why the intersection only has to be finite? I was wondering if there is a reason why in the definition of a topologic space, the reunion of any collection of open sets is also an open set but the intersection has to be finite to be a set.  Is there because of the definition of an open set itself ? I have always seen it as a definition, but now I want to know why.  Thanks for your help. I need to learn basic optics for a high school academic competition. Does anyone know any good places, preferably free and online, to learn the basics of optics, like lenses, angles of incidence, etc.? I can't seem to find an MIT OpenCourseWare course addressing the issue. Occasionally, I've come across a webpage that tries to pop open a new window (for user input, or something important), but the popup blocker prevents this from happening.  What methods can the calling window use to make sure the new window launched properly? If $A = \tan6^{\circ} \tan42^{\circ},~~B = \cot 66^{\circ} \cot78^{\circ}$ find the relation between $A$ and $B$ My trigonometric problem is:     If $A = \tan6^{\circ} \tan42^{\circ}$ B = cot$66^{\circ} \cot78^{\circ}$ find the relation between $A$ and $B$.    Working :   $$B = \cot 66^{\circ} \cot78^{\circ} = 1- \frac{\tan24^{\circ}+\tan18^{\circ}}{\tan42^{\circ}}$$  $$A= \tan6^{\circ} \tan42^{\circ} = 1- \frac{\tan6^{\circ} +\tan42^{\circ}}{\tan48^{\circ}}$$  but it seems  this is the wrong way of doing this...please suggest. Thanks! Why don't we treat rubbish the same as spam? Why is the subject omitted in sentences like "Thought you'd never ask"? "Thought you'd never ask" is "I thought you'd never ask" with "I" omitted. "Hope this helps" is "I hope this helps" with "I" omitted.   In English grammar, normally every sentence should have a subject, right?   My first thought is that these two examples are so often used that they are like set phrases. But these are not really set phrases. You can alter the words after "thought" and "hope".  Another possible explanation is the tendency to drop the subject if it is the first person pronoun. It seems that in many languages, such as Spanish, Italian and Japanese, the first person subject is usually omitted. Maybe English is going the same way? (Not exactly the same, since in Italian, verb forms change according to the person, so the subject is not necessary to understand who one's referring to.)  And, apparently, such omission is more common in spoken English than in written English.  Are there more examples of such first person subject omission? How frequent is it? Strange arrow mark with TikZ edge and anchors. When I draw an edge between two nodes with anchors and globally set option [-&gt;], I get an additional arrow tip symbol at the beginning of the last edge in the path. For example in  \documentclass{article} \usepackage{tikz}  \begin{document} \begin{tikzpicture}     \node (A) at (0,0) {$A$};     \node (B) at (1,0) {$B$};      \draw[-&gt;] (A.north east) edge (B.north west);      \draw (A.south east) edge[-&gt;] (B.south west); \end{tikzpicture} \end{document}   the first \draw command produces incorrect output, while the second produces correct output:    Am I doing something wrong, or is this a bug in TikZ?