How to calculate the value of the limit of a sum: $\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\frac{k}{k^{2}+n^{2}}$? Need to calculate the value of the following limit $$\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\frac{k}{k^{2}+n^{2}}$$ I don't know how. In general would know how to make that kind of limits with summations, where can I find examples and learn the topic, I appreciate any hint to make this limit.
How do you calculate this limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k}{n^2+k^2}$? 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--  $\begin{align} &amp;A) \dfrac{\pi}{2} \\ &amp;B) \log 2 \\ &amp;C) \dfrac{\pi}{4} \\ &amp;D) \dfrac{1}{2} \log 2 \end{align}$
How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical community's standards? I'm a computer science student who is a maths hobbyist. I'm convinced that I've proven a major conjecture. The problem lies in that I've never published anything before and am not a mathematician by profession. Knowing full well that my proof may be fallacious, erroneous, or simply lacking mathematical formality, what advice would you give me? How do I disable touchpad when using a mouse Referencing bib files with spaces in the filename I am using a Mac with TeX Live and Mendeley for automatically generating .bib files, some of which have spaces in the file names. I would like to know how to reference such files from LaTeX as using "" or \ doesn't seem to work. This is what my  looks like when it first loads:    But if we unselect math.SE, we get a y-axis rescaling mess:    I'm using Firefox. I'm in China, so many things involving Google and Imgur are blocked, and I use  to stop the browser attempting to load unloadable Google objects.  I'm hoping other users can check whether or not this issue is localized to my system, and if it's not just me, here's the bug report. Why doesn't a bike/bicycle fall if going with a high speed? How (and why) use stderr for both reading and writing? According to , less reads navigation commands from stderr if it's not able to open /dev/tty.  This seems puzzling, since I've never seen anything write to another program's stderr stream, and I don't know how I would even accomplish that.  What is the purpose of stderr being open for both reading and writing? And if this is useful, how do I make use of it on modern systems? (Is there some arcane syntax to pipe something into stderr instead of stdin, for instance?) Take a look at this example schematic:  The chip in question is the  and as you can see power is provided on both sides (with a 100nF cap to smooth noise I guess). Is this strictly required or could you put power into one side and just ignore the other? I know that I have just put power in one side and everything still seems to work fine but I feel uneasy with doing that. I feel that they would not waste pins that could be doing something useful by having an extra pair of power in/out pins. For what purpose does this PIC MCU (and others like it) have multiple power pins? I am new to photography, and need a little advice. I currently have a .   To get rolling, where do I go from here? Does the camera matter? Is this camera enough to start with? How do I pick a part that goes into a breadboard when I order it? I have just bought some 555 timers and they are timers for microelectronic circuits...could anyone tell me what I need to search for in order to get larger ones which will fit a breadboard? Thanks Is there a LaTeX/TeX preprocessor/inliner that replaces only selected macros with their definitions? What can I do to lift a 10 year ban for making a mistake in my UK visa application? I am a Nigerian citizen.  In 2007 I was twice denied a visa to the UK.  Recently I was filling out an application for another UK visitor visa and by mistake I answered no to the question whether I had ever been denied a UK visa, instead of yes.  When reviewing my application, the visa officer found out about the previous visa rejections and as a result I got a 10 year ban on travel to the UK.  What can I do to lift my 10 year ban?   Will this ban affect my chances of obtaining a visitor visa to the USA? how can i forward requests to my own IP address to a local IP address in my network (possibly using named)? My setup  I have a linux server running in my basement. I currently have my router looking to that server for DNS lookup, so I can use named to spoof a bunch of domain names to point to my local server's local address, 192.168.0.111. This works great.  I also am running Apache and PHP on this server, and have forwarded port 80 on my router to this machine, so my public IP xx.xxx.x.xx is sent to the basement server.  This is a great development set up for me, because I can develop websites in my local network and pull them up on numerous devices with a *.mylocal.net address. I can also quickly share a website on my public IP with others that are outside of my network.  I have a static IP, so that makes this even easier.  The problem  Some applications need to have an absolute URL. My spoofed DNS doesn't work outside my network, and neither does my internal IP address.  I need to be able to access the sites at the same IP that they are accessed publicly, which is my public IP: xx.xxx.x.xx    For some reason, I can't access my own public IP address from within my own network.  My router's public IP is xx.xxx.x.xx, and when I hit xx.xxx.x.xx from outside my network I see exactly what I want -- my development website. But, when I hit xx.xxx.x.xx from inside my network, the connection times out.    I assume my ISP has a firewall that blocks requests from xx.xxx.x.xx to xx.xxx.x.xx where the origin and the destination are the same IP (or maybe this is impossible on any network?)  So how can I set up my network to forward the internal request to my public IP to the local IP of my basement server?   I would like to do this within the local DNS server I set up, or in my router, so the change can be applied to all devices in my network, and not just the basement server or the windows tablet/laptop, etc individually. How is light affected by gravity? Google AJAX API - How do I get more than 4 Results? How to remove Home Folder icon from desktop in Ubuntu 20.04? When I start Ubuntu I had my &quot;home/user&quot; in my Desktop. test is my User.   How do I make the folder not be on the desktop? In java SSL over https without certificate Using the expression "the same" for a previously mentioned item I frequently use &quot;the same&quot; as in the following sentence.  I need the details of XYZ project. Could you please send me the same as soon as possible?  Is this grammatically correct? I have not been corrected till now, but I have not seen the usage from native speakers very much either. Update: Seems like my example sentence was ambiguous. Hope the following sentence is better.  I need the documents for the meeting. Could you please send me the same as soon as possible? My 12.04 x64 system won't boot into GUI, but I can still use virtual console with my own account, so apparently it's the Windows Management System that's causing it, how can I reinstall it from tty, what's the apt-get command? I want to put the section title in the left margin using titlesec. However, it results to a superfluous empty line when using list environment. How should I remove it without using vspace?  Minimal example:  \documentclass{article}  \usepackage{geometry} \geometry{   a4paper,   top=2cm,   bottom=2cm,   left=4.5cm,   right=1.5cm, }  \usepackage{titlesec} \titleformat{\section}[leftmargin]{\large\scshape\sffamily}{}{0cm}{} \titlespacing{\section}{2.5cm}{2ex}{0.5cm}  \usepackage{enumitem}  \usepackage{mwe}  \begin{document}  \section{Section A}  This line is aligned with the section title.  \blindtext  \section{Section B}  \begin{itemize}[nosep]   \item This line is not aligned with the section title. \end{itemize}  \end{document} Gauss's mean value theorem problem Suppose that  $f(z)$ is analytic and $f(z)\neq0$  on the disc $|z|&lt;1$ .  Show that for $0&lt;r&lt;1$,  $$ \exp\left[\frac{1}{2\pi}\int_{0}^{2\pi}{\log}|f(re^{i\theta})|\,d\theta\right] = |f(0)| $$  Any other theorems needed to solve this problem? I don't have any idea just with "Gauss's MVT"  Thank you in advanced.