Find the limit by expressing it as a definite integral of an appropiate function via Riemann Sums. Find the limit $$\lim_{n\to \infty}\sum_{i=1}^{n}\frac{i}{n^2+i^2}$$ by expressing it as a definite integral of an appropiate function via Riemann Sums.  Observation: $n$ must refer to the number of slices, and $i$ must refer to $i$th slice. My attempt. First I revisited Riemann Sums. Assume what I am trying to find have the form $\int_{a}^{b}f(x).$ Cutting up the bound $(a,b)$ into $n$ slices, the length of each piece with respect to $x$ is $\frac{b-a}{n}. $ Next, looking at each slice, I decide to take the right hand value for convenience, that is $a+i\frac{b-a}{n}. $ Now, I clearly have the area of each slice, that is, $$\frac{b-a}{n}*f(a+i\frac{b-a}{n} )$$ And when I sum up all of the slices, I have $$\sum_{i=1}^{n}\frac{b-a}{n}*f(a+i\frac{b-a}{n} )$$ And finally, increasing the number of cuts to make the area as accurate as possible, we have $$\lim_{n\to \infty}\sum_{i=1}^{n}\frac{b-a}{n}*f(a+i\frac{b-a}{n} )$$ Therefore I conclude that $$\frac{i}{n^2+i^2}=\frac{b-a}{n}*f(a+i\frac{b-a}{n} )$$ Whats left now is to find b,a and f(x). After all the work, I feel closer to my answer, yet so far away from it.  Any hints? Thanks in advance! List them as solutions. I am looking for hints only.
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--  $\begin{align} &amp;A) \dfrac{\pi}{2} \\ &amp;B) \log 2 \\ &amp;C) \dfrac{\pi}{4} \\ &amp;D) \dfrac{1}{2} \log 2 \end{align}$
Difficult Polynomial Question Let $\ln X \sim \operatorname{Exp}(1)$ and $X,Y$ i.i.d.  What is the distribution of $Z=XY$?  My attempt: $$P(Z\le z)=P(XY\le z)=P(\ln(XY)\le \ln(z))=P(\ln(X)+\ln(Y)\le \ln(z))$$  We need the convolution, since $\ln X $ and $\ln Y$ are independent we get, $$\int_0^{\ln(z)} f_{\ln(X)}\circledast f_{\ln(Y)}=\int_0^{\ln(z)}e^{-y}e^{-x+y}\,dy=e^{-x}ln(z),$$ i.e. $e^{-x}\ln(z)$ is the density of $XY$ Is there a page listing all fields? i have a scenario where i want to know in which position number the column name comes   like i have sample data below   ID|NAME|S1|BQ|S2|VN|D1 1|vimal|10.12|12.4|56.2|12.2|11.22 2|vilas|10.12|12.4|56.2|12.2|11.22 3|viky|10.12|12.4|56.2|12.2|11.22   Colum name : S1 S2 D1   output should be :  3 5 7 What's the easiest way to unrar a file? Let $f : \mathbb R^n \rightarrow \mathbb R$ be a measurable function.  We know that $f$ is not necessarily locally integrable - but how hard can a measurable $f$ fail to be so? For example, is it true that $f$ is locally integrable outside of a negligible set $N \subset \mathbb R^n$?   Edit: In consideration of the comment by zhs, let me correct the phrasing of the question. I was originally wondering whether the 'singular' behaviour of a measurable function can be localized to a 'small' set.   A motivational example is $f(x) = 1/x$ over the real line. It is locally integrable in the complement of the origin. Hence, a conceivable phenomenon, which I had in mind originally when asking the question, could be that a measurable $f$ is integrable over open sets that are compactly contained in the complement of some negligible sets. (used to show two posts from MSE)   (there used to be a list of the accounts)   (it was there) seems to be fixed I have to resit a calculus exam and for some reason set proofs were never my best friend...  Anyway, on a practice exam I encountered the following proof:  $$A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$$  When I draw a Venn-diagram it seems quite obvious but I couldn't manage to write the proof down properly.  If someone could help me, that'd be great! How can a quasar be 29 billion light-years away from Earth if Big Bang happened only 13.8 billion years ago? I was reading through the Wikipedia article on  and came across the fact that the most distant Quasar is 29 Billion Light years. This is what the  exactly says      The highest redshift quasar known (as of June 2011[update]) is ULAS-J1120+0641, with a redshift of 7.085, which corresponds to a proper distance of approximately 29 billion light-years from Earth.   Now I come to understand that the  singularity is believed to be around 13.8 Billion years ago.  So how is this possible? Does the presence of such a quasar negate the Big Bang Theory?   I'm not a student of Physics and was reading this out of (whimsical) curiosity. Is there something I'm missing here or the "proper distance" mentioned in the fact is a concept that explains this?  Edit: My Bad! Here's how..  A simple google search led me to  which says the farthest quasar found is 12.9 billion LYs and not 29 billion. So in the end we have just proven that wikipedia needs more moderation. Script-r Symbol The near-ubiquitous undergraduate reference on electrodynamics, "Introduction to Electrodynamics" by David Griffiths make extensive use of a script-r symbol, defined to be \vector{\scriptr} \equiv \vector{r} - \vector{r}^{\prime} and looks like:    Although the question of how to produce such a symbol in LaTeX is quite a common one I am yet to find a satisfactory solution.  The closest I've found is from the physymb package and uses the Calligra fonts:    Sadly, there is not a bold variant of this character and it is far more cursive than I would like.  I am hence wondering if there are any better facsimiles. Stacks for Unity Launcher I'm looking forward to emulate the behavior of OS X Dock Stacks so that I can have, for instance, one stack for browsers and another one for IDES. People at ask ubuntu are doing a lot of cleaver stuff with custom Launchers and Quicklists (see ). So has anyone figured out how to have stacks in Unity? Anything fancier than a static Quicklist over a dummy folder icon? What should be in the sources.list for Kali Rolling Linux? This is exact content from my sources.list:  #  # deb cdrom:[Debian GNU/Linux 2016.1 _Kali-rolling_ - Official Snapshot amd64 LIVE/INSTALL Binary 20160830-11:29]/ kali-rolling contrib main non-free #deb cdrom:[Debian GNU/Linux 2016.1 _Kali-rolling_ - Official Snapshot amd64 LIVE/INSTALL Binary 20160830-11:29]/ kali-rolling contrib main non-free   I couldn't install the Synaptic package manager and was told to modify sources.list.  I googled a lot and added some repos into it and ran update and upgrade command only to see Kali crash... I had to recover it.  Please mention the exact content that should be there to ensure that Kali Linux works. I've just installed Ubuntu 14.04 and am trying to install php 5.3. But when I run   apt-get install php5    it's installing php 5.5.9-1ubuntu4. I need to use php 5.3 in the system because the application I'm working on is built with php 5.3.   How can I achieve this? How can I return pivot table output in MySQL? Can't paste data to question, get prompted for Image? I am aware of the basics of evolutionary theory, however I don't understand how mutations can add genes over time.   Am I correct in thinking that creatures within the same species who mutate to have an additional gene in their genome would normally be infertile?  Or am I misunderstanding that?  Can someone explain the process of the creation of genetic material through evolution?  Citation to a decent academic paper or book on the matter would be appreciated too. How can I quickly level up in World of Warcraft? I've been playing for months and I'm barely at level 40, while my classmates play less than I do and have multiple lvl 80 characters.  Is there a better way to level fast in World of Warcraft than just completing quests? How can I `alias sudo !!`? I'm trying to set an alias for sudo !! in Bash. I tried alias sbb='sudo !!', but it interprets that as a literal  !! and prints   sudo: !!: command not found   If I use double quotes, it substitutes the double bang in the string itself, so that doesn't work.  Is there any way to make this work? Or an alternative alias? ` Can you explain closures (as they relate to Python)? I've been reading a lot about closures and I think I understand them, but without clouding the picture for myself and others, I am hoping someone can explain closures as succinctly and clearly as possible.  I'm looking for a simple explanation that might help me understand where and why I would want to use them. Watch Filesystem in Real Time on OS X and Ubuntu I'm looking for a CLI tool which will watch a directory and spit out the names of files that change in real time.   some_watch_command /path/to/some/folder | xargs some_callback   I'm aware of inotify (inotify-tools?) and it seems to be what I need, but I need something that is both Linux (in my case Ubuntu) and OSX compatible.  It doesn't need to be lightning fast, but it does need to trigger upon changes (within a second is reasonable). Also, I don't necessarily need the exact CLI program mentioned above.  If some underlying tech exists and is easily scriptable on both platforms that would be great too.