Limit of the sequence $\frac{1^k+2^k+...+n^k}{n^{k+1}}$
Evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}}$
Why does demand for stock rise when a company appears to have high future value? Allow suppression of automatic linking of web addresses How to find a generator of a cyclic group? Does lower resolution result in better signal-to-noise ratio, given the same sensor size? With the release of the Canon EOS R5/R6, there have been many saying that they prefer the 20MP R6 camera over the 45MP R5 camera, due to its lower price but also because of &quot;better low-light performance.&quot; In camera comparison sites, I've also seen the sentiment that bigger pixels (lower resolution) = better low light performance. However, is this a myth? Here is my reasoning for this. Let's say that there are two full-frame sensors, one 20MP sensor and a 80MP sensor. According to many people, the 80MP sensor would be terrible for low-light photography since the pixels are smaller. However, each 2x2 block in the 80MP sensor could be mapped, and in post averaged to a 1x1 pixel, effectively making the image into a 20MP image. This could also be done in-camera as an option. Would this image have just the same amount of usable information as the image from the 20MP sensor? In other terms, since both sensors are 35mm sensors (i.e. equal area), the amount of light falling onto the sensor is the same. So, at the end of the day, there will be no difference between a 20MP image and a 20MP image that has been downsampled from 80MP, right? So their low-light performance would be the same. You even get an advantage from having the option to get 80MP photos if you wanted. Why do people say that a lower-resolution camera has better low-light performance, when it should theoretically be the same as one with higher resolution (all other factors equal)? Note: this question is about sensors with the same size but with different pixel density. Hence, it is different from  asking about different sensor sizes. I am aware that a full-frame camera will have better low-light signal due to its bigger area than an APS-C sensor, but what about the similarly sized sensors? Word for person who looks for flattery I remember coming across this fact a while ago in a pdf somewhere, but I haven't been able to find it again. Can someone show me how to prove it? I would appreciate easier proofs.  EDIT: I'm very sorry, I forgot to actually state it.  Prove that $-1$ is not a quadratic residue modulo primes of the form $4k+3$.  Thanks! "approach to address" or "approach to addressing"? Depending on our point of view, we offer different approaches to address(ing?) a same issue.   Should I use &mdash;ing in this context, and if yes, why? I have a symlink with these permissions:  lrwxrwxrwx 1 myuser myuser       38 Aug 18 00:36 npm -&gt; ../lib/node_modules/npm/bin/npm-cli.js*   The symlink is located in a .tar.gz archive. Now when I unpack the tar.gz archive using maven the symlink is no longer valid. I'm therefore trying to reconstruct the symlink. First I create the symlink using ln but how do I set the same permissions as the original symlink? a) Let $a&gt;0$ and the sequence $x_n$ fulfills $x_1&gt;0$ and $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$ for $n \in \mathbb N$. Show that $x_n \rightarrow \sqrt a$ when $n\rightarrow \infty$. I have done it in two ways, but I guess I'm not allowed to use the first one and the second one is incomplete. Can someone please help me?  We already know $x_n \rightarrow \sqrt a$, so we do another step of the iteration and see that $x_{n+1} = \sqrt a$.  Using limit, $x_n \rightarrow x, x_{n+1} \rightarrow x$ (this is the part I think it's incomplete, don't I have to show $x_{n+1} \rightarrow x$, how?), we have that   $$x = \frac x 2 (1 + \frac a {x^2}) \Rightarrow 1 = a/x^2 \Rightarrow x = \sqrt a$$ b) Let the sequence $x_n$ be defined as $x_{n+1}= 1 + \frac 1 {x_n} (n \in \mathbb N), x_1=1$. Show that it converges and calculate its limit. &quot;Tip: Show that sequences $x_{2n}$ and $x_{2n+1}$ monotone convergent to the limit.&quot; I didn't understand the tip, how can this help me? Does it make a difference if the number is odd or even? Thanks in advance! Command and Conquer 3 exeption in exception handler I have command and conquer 3 from Origin, but when I try to start it, I get the 'exception in exception handler' error and the game exits.   I've tried the Origin launcher fix as well as -runver to make the game run a different version (although this just says that it can't find any version). I run Windows 10 (I know, experimental) but this game worked at one point, and then stopped working suddenly - there wasn't even an update in between.   I suspect that it's got to do with Origin, but any help would be appreciated. How can I stop mobs from despawning? I am using Spigot, some plugins, and I want to make like a zoo on a map, with mobs behind glass cubes. The problem is that when I load my map, every time some of the mobs just vanish, empty cages. I put eggs again, they spawn, I go to my other map in the multiverse (/mv tp my_other_map) and when I come back, empty cages again, always the same ones...   I've  read it's a question of distance of me (distance in cubes unit), if I'm too far away, the mobs will despawn... Is there a way for them to stay? Possessive 's for things "[" as first character in table row Whenever I try to compile a table that has a "[" as first character in a row it gives me the error:  Missing number, treated as zero. Illegal unit of measure   Example:  \begin{table}[htb]     \centering     \begin{tabular}{r|r}         [some text ] &amp; ... \\         [some more text ] &amp; ...     \end{tabular}     \caption{...}     \label{tab:...} \end{table}   How can I print "[" as first character in a row? I have a doubt on something. I have a theorem that says that if $R$ is a ring, then $R/I$ is a domain iff $I$ is prime. Since $X^2-1$ is note prime, $\mathbb Z[X]/(X^2-1)$ is not a domain. But, $X^2-1=(X-1)(X+1)$, and since $X-1$ and $X+1$ are prime in $\mathbb Z[X]$, there are coprime, and thus, by the chinese remainder $$\mathbb Z[X]/(X^2-1)\cong \mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1),$$ since $\mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1)$ is a domain, then so is $\mathbb Z[X/(X^2-1)$. What is the mistakes here ? We all know very well how to make an angle of 15 and 45 degree and its multiple using compass. Can anybody tell me how to make an angle of 10 degree and its multiple using compass? Why is '.' a hard link in Unix? How often did Gandalf return to Hobbiton after the events of The Hobbit? Proving that $ 2 $ is the only real solution of $ 3^x+4^x=5^x $ I would like to prove that the equation $ 3^x+4^x=5^x $ has only one real solution ($x=2$)  I tried to study the function $ f(x)=5^x-4^x-3^x $ (in order to use the intermediate value theorem) but I am not able to find the sign of $ f&#39;(x)= \ln(5)\times5^x-\ln(4)\times4^x-\ln(3)\times3^x $ and I can't see any other method to solve this exercise... What is the 'rev' dropdown supposed to do? I only see this on sites where I don't have edit privileges, and only where there have been multiple revisions:        What exactly is this supposed to do?  Intuitively, I would expect it to change the text in the box to what it was at that revision, but it doesn't. And, if that's it, why only where I need to suggest edits? How to make sure I'm not using any proprietary software after installation? I know I can check the currently installed non-free software with the following command (assuming aptitude is FOSS):  aptitude search '?installed (?section(restricted) | ?section(multiverse))'   And I can remove the "restricted" and "multiverse" sources (and not add any PPA's) to keep that list empty. I'm assuming from this point, whenever I apt install something, that something is either in the allowed sources or it is not found.  Is this enough? Does this make sure all software I'm running (in Ubuntu) is open-source?