I guess it has something to do with Riemann sums but this is new for me.  $\displaystyle\lim \limits_{n \to \infty}\sum \limits_{k=n}^{2n}\frac{k}{k^2+n^2}$  How do I start?
How to find the value of $\lim_{n\to\infty}S(n)$, where $S(n)$ is given by $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$   is unable to calculate it.  This is a question from a questions booklet, and the options for the answer are--  $\begin{align} &amp;A) \dfrac{\pi}{2} \\ &amp;B) \log 2 \\ &amp;C) \dfrac{\pi}{4} \\ &amp;D) \dfrac{1}{2} \log 2 \end{align}$
Access .mat file containing matlab classes in python Beamer: bold math no longer working Creating point features with exact coordinates in QGIS Print LIST of unicode chars without escape characters The residue at $\infty$ Can you have multiple Dropbox accounts on one computer? Is Using Digital Signatures to prove identity a zero knowledge proof? Suppose Alice publishes a book with a public key in it, and later wants to prove that she wrote the book. She could sign challenge messages with her private key, and others could verify those signed messages. Alice is proving she knows a private key without revealing anything about it.  Is this an example of an interactive zero knowledge proof?   Or does this example somehow not apply to zero knowledge proofs? Do I keep my Windows installation if I install Ubuntu with Wubi, the Windows installer? Solve $\frac1x+\frac1y=\frac1{pq}$ For $x,y\in\mathbb{N}$ how many ordered pairs $\left(x,y\right)$ satisfy   $\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{pq}$ where $p,q$ are distinct primes? Initializing Manipulator value within DialogInput (version 10.1 only) There is a command l available on my machine which appears to do nothing.  which l also produces no output. Is this a real command, and does it actually do anything? $\infty = -1 $ paradox I puzzled two high school Pre-calc math teachers today with a little proof (maybe not) I found a couple years ago that infinity is equal to -1:   Let x equal the geometric series: $1 + 2 + 4 + 8 + 16 \ldots$  $x = 1 + 2 + 4 + 8 + 16 \ldots$ Multiply each side by 2:  $2x = 2 + 4 + 8 + 16 + 32 \ldots$ Again from the equation in step 1, move the $1$ term to the left hand of the equation:  $x - 1 = 2 + 4 + 8 + 16 + 32 \ldots$ So the following appears to be true:  $2x = x - 1 \implies x = -1$   This is obviously illogical. The teachers told me the problem has to do with adding the two infinite geometric series, but they weren't positive. I'm currently in Pre-calc, so I have extremely little knowledge on calculus, but a little help with this paradox would be appreciated. Legal issues for a "fangame" of a commercial franchise? I've played around in the past of the 2D Flash fan-made version of the popular Valve game Portal.  It has basically the exact same mechanics, but as a 2D side-view flash game.  Lately I've toyed around with creating a simplified flash version of some of my favorite franchises (such as Zelda, Mario, Splinter Cell, etc.).  Of course, nothing is "stopping" me from just using the same mechanics and just changing the name to something not copyrighted.  I have witnessed projects that have tried to directly port a 3D game into a 2D simplified version, with the exact same plot and puzzles (Ocarina of Time in 2D comes to mind), and I have seen them being shut down.  But I am not likely going to exactly duplicate any game...just borrow some of the source material and basic underlying mechanics.  For example, if I were to make a Zelda-influenced fan game, I would have items such as the Hookshot, Boomerang, and all other sorts of ones from the Gameboy Zeldas; just with a custom plot and dungeons.  With likely the same mechanics.  What are the legal issues involved in this?  I used to think it was a "Don't do this ever; it's a bad idea."  But then I've seen the Portal clone, and now I'm curious. I'm using a docker image as a base for my own development that adds the jessie backports repository in its Dockerfile and uses that to install a dependency. This image uses the following command to add the repository:  echo "deb http://ftp.debian.org/debian jessie-backports main" &gt;&gt; /etc/apt/sources.list   The problem is that fetching packages from the backports repository now  fails with the following error (this used to work previously):  W: Failed to fetch http://ftp.debian.org/debian/dists/jessie-backports/main/binary-amd64/Packages 404  Not Found  W: Failed to fetch http://deb.debian.org/debian/dists/jessie-updates/main/binary-amd64/Packages  404  Not Found   I looked on that server, and those paths are indeed not present there.  I tried to figure out on the Debian backports site whether this particular repository should still be available, and I didn't find any indication that this was deprecated or something like that.  Is this a temporary issue with the repository, or is the jessie-backports repository not available anymore? And if this is not a temporary issue, what options do I have to use this or an equivalent repository without upgrading to the newer Debian stable version? How to render a background image in 2.8? I'm trying to "add" an 3D object onto a photo. I have added the photo to the scene in three ways so far, but none of them render (eevee):   Camera > Background Images Empty a. Add > Empty > Image (Empty gets added to current Collection) b. Add > Image > Backround (Empty gets added to Scene Collection)   These all show up in the 3D View and I was able to align the camera and the object so its perspective matches the scene in the photograph. But when I Render the scene, I only see the object, not the background image.  For the latter, it's simple: Empties are not rendered at all. There are few questions about this, since 2.8 has changed how it handles background images. The most popular seems this one (it includes references to others):      But I seem not to be able to find a method to have the background image render.  Another method might be to use the Compositor (Nodes) to Mix or Alpha Over an Image with the Render Layers. But this does not give the same alignment as achieved with the background image (also this requires that the World background does not get rendered (Alpha channel); for Eevee, this can be achieved by checking Film>Transparent).  What other options are there?  Docs: proof of functional completeness of logical operators Different mechanics of \hyperlink vs. \hyperref I am fiddling around with the hyperref package to build a hypertext structure into my pdf. At first glance, hyperref provides two similar macros to put links between parts of my document: \hyperlink{target}{text} and \hyperref[target]{text}. You might notice their slightly different syntax. What really interests me is their different working mechanics. I will illustrate this by the following example: \documentclass{article}  \usepackage{pdfpages} \usepackage[]{hyperref} \usepackage{mwe}  \begin{document}  \section{Section Heading} \label{sectionheading}  \includepdf[link=true, linkname=pdfpage]{example-image}   Link to section works with \hyperref[sectionheading]{hyperref} and with \hyperlink{sectionheading}{hyperlink} (but with slightly different targets on the page).  Link to pdf works with \hyperlink{pdfpage.1}{hyperlink} but not with \hyperref[pdfpage.1]{hyperref} (and not this way, either: \hyperref[pdfpage]{hyperref}).  \end{document}  To link to the labeled section heading, I may use both commands. While \hyperref directs my pdf viewer directly to the linked section, \hyperlink directs me somewhere slightly different, so I can see more context. To link to the \includepdf of pdfpages however, I may only use \hyperlink. \hyperref does not work at all. I suspect this is because of a subtle difference in their working mechanisms. While checking the aux file, I found a line \newlabel{sectionheading}{{1}{1}{Section Heading}{section.1}{}} which \hyperref seems to use as a target. However, I did not find any similar target for the included pdf. My question: I suspect, I understand the workings of \label and \ref very well. \hyperref seems to work quite similarly with the added clickability. Could you please explain to me how \hyperlink does work differently and how it is implemented? E.g. what mechanics explain the difference when clicked on the link and where and how are the link targets stored in the first place? How to find the sum of all the numbers that can be formed using the digits 2,3,3,4,4,4.. What should be the way of doing this type of problem?? Please guide Factorisations of permutations I'm trying to use itemize inside a tabularx environment, but I have a problem when I mention the end of the line. It seems that using \\ end the line after the end of itemize, resulting in a supplementary vertical spacing that I would like to remove.  Any hint?  \begin{frame} \frametitle{Sujet de thèse}         \begin{center}             \textit{Sujet N$^\circ$221 : Formation et dynamique des champs de dunes longitudinales.}         \end{center}          \begin{tabularx}{\textwidth}{l !{\color{blue}\vline width 3pt} X}             Contexte &amp;  \scriptsize{Classification des dunes selon leur mécanisme de croissance, leur orientation et leur forme.              Bonne compréhension :             \begin{itemize}[noitemsep,topsep=0pt, label=\textbullet]                 \item du mécanisme de croissance lié à une disponibilité en sédiment grande.                 \item de la croissance sous un vent unidirectionnel.             \end{itemize}} \\             Objectifs &amp; \scriptsize{Etude de l'orientation longitudinale :             \begin{itemize}[noitemsep,topsep=0pt, label=\textbullet, parsep=0pt ]                 \setlength\itemsep{0cm}                 \item Développement d'un formalisme sous des vents multidirectionnels.                 \item Etude des dunes longitudinales dans le cas d'une faible disponibilité en sédiment. (Fingering mode)                 \item Inversion: données topographiques $\longrightarrow$ Données de vents $\rightarrow$ paléo-climats.              \end{itemize}} \\             Outils &amp; \scriptsize{Modèle numérique (Rescal), Données expérimentales, Théorie.} \\         \end{tabularx}     \end{frame}     And yes, dunno if it's important but everything is inside beamer.