Evaluate $$\lim_{n\to \infty}\frac{n+n^2+n^3+....+n^n}{1^n+2^n+3^n+....+n^n}$$  My Attempt  Was able to write as $$\lim_{n\to \infty}\frac{\frac{n^n-1}{n-1}}{n^n(\frac{1}{n})\sum_{r=1}^n\left(\frac{r}{n}\right)^n}$$  But not sure how to take it further though the denominator appears to be Riemann Sum
What is the limit of:  $$\lim_{n\to \infty} \frac{n^1+\dots+n^n}{1^n+\dots+n^n} = ?$$  I did some computation with big numbers, I guess it is in the interval $\left(\frac{1}{2},1\right)$.
If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show that $\lim _{x\to \infty } (f(x+1)-f(x))=0$. Having some trouble proving this:     If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show: $\lim _{x\to \infty } (f(x+1)-f(x))=0$   Attempt: from $\int \lim _{x\rightarrow \infty } f^\prime(x) \mathrm{d}x=0$ we can say that $f^\prime(x)=const$ would we then have to show that  $\lim _{x\to \infty } (f(x+1))$ is also a constant? From there if we show that both are equal then the difference is zero. I am on the right track?  Thank you for the help. Why am I getting "line 1: $' :\r': command not found"? I use Cygwin on my laptop (DOS). I have a collection of scripts from my colleagues, and my own. I am not an IT person, not knowledgeable in Unix. I am following my colleagues' syntax and able to manage a few simple things. The scripts worked well on my old laptop. I just changed laptop and installed Cygwin. When I run my scripts, they do not work. Here is one example of the error message I get: line 1: $':\r': command not found line 5: syntax error near unexpected token `$'\r'' line 5: `fi  Here are the top 5 lines of my script : iter=1 if [ -f iter.txt ]    then rm ./iter.txt  fi    Can someone please explain how I can get around this problem? When I cd a link, my current path is prefixed with the link's path, rather than the path of the directory the link links to.  E.g.  ~/dirlinks/maths$ ls -l logic lrwxrwxrwx 1 tim tim 71 Jul 27 10:24 logic -&gt; /windows-d/academic discipline/study objects/areas/formal systems/logic  ~/dirlinks/maths$ cd logic ~/dirlinks/maths/logic$ pwd /home/tim/dirlinks/maths/logic  ~/dirlinks/maths/logic$ cd .. ~/dirlinks/maths$   I would like to have my current path changed to the path of the linked dir, so that I can work with the parent dirs of the linked dir as well. Besides ls the link to find out the linked dir, and then cd into it, what are some simpler ways to accomplish that? For example, after cd into a link, how do you change your current path to the path of the linked dir? That is, what are some good examples of vector spaces which are inner product spaces but in which not every Cauchy sequence converges? What is the definition of $2.5!$? (2.5 factorial) I was messing around with my TI-84 Plus Silver Edition calculator and discovered that it will actually give me values when taking the factorial of any number $n/2$ where $n$ is any integer greater than $-2$. Why does this happen? I thought factorials were only defined for positive integers and $0$, so what is my calculator doing to get the answer $3.32335097$ when I enter $2.5!$? Is there actually a definition of $2.5!$ or is my calculator just being weird? How is the factorial function implemented?  I understand the binary implications of $2.5$, so that could possibly have something to do with it. I get a domain error when trying to take the factorial of $-1, 2.3, e$, and any number that is not of the form $n/2$ where $n$ is any integer greater than $-2$. Is there actually a 0 probability of finding an electron in an orbital node? I'm trying to fill out the skills section on my first character, but when I look at the character sheets in the Starter Set as examples of how to make a character, the skills that are circled have higher modifiers than their Dexterity or Strength or whatever should make them.  I've filled out all skills but those. What do I add to the ability modifier to get those numbers?   My abilities are Strength 13, Dexterity 14, Constitution 11, Intelligence 15, Wisdom 8, and Charisma 8. Proficiencies are Arcana, Athletics, History, Investigation and Stealth. My saving throws are Strength and Dexterity.   I just started D&amp;D and have the Player's Handbook, DM's Guide, Monster Manual, and the Starter Set. "Reality" of length contraction in SR Maclaurin series of $\ln(1-x+x^2)$ Is there a closed form expression for $\text{n}^{\text{th}}$ coefficient of Maclaurin series of  $$\ln(1-x+x^2)$$ Javascipt component not coming render as pdf Is JavaScript's "new" keyword considered harmful? In another , a user pointed out that the new keyword was dangerous to use and proposed a solution to object creation that did not use new. I didn't believe that was true, mostly because I've used Prototype, Scriptaculous and other excellent JavaScript libraries, and everyone of them used the new keyword.  In spite of that, yesterday I was watching Douglas Crockford's talk at YUI theater and he said the exactly same thing, that he didn't use the new keyword anymore in his code ().  Is it 'bad' to use the new keyword? What are the advantages and disadvantages of using it? What do the .eh_frame and .eh_frame_hdr sections store, exactly? What are the rules for calling the base class constructor? Where is the default folder for Apps? I am relatively new to the linux world. I use OSX at home and Windows at work. In those worlds, there is a default folder for applications (C:\Program Files, /Applications). Is there any such directory in Ubuntu?   I understand that this is not very important and any location in the file system should be ok, but was just wondering whether there are such standard locations (I think there would be one). For instance where is Firefox or Libre Office (that are installed by default)?  Got this question when I was trying to install eclipse. Downloaded the zip, and was wondering... Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$ I was hoping to find a more "mathematical" proof, instead of proving logically $\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$.  I already know the logical Proof:  $${n \choose k}^2 = {n \choose k}{ n \choose n-k}$$  Hence summation can be expressed as:  $$\binom{n}{0}\binom{n}{n} + \binom{n}{1}\binom{n}{n-1} + \cdots + \binom{n}{n}\binom{n}{0}$$  One can think of it as choosing $n$ people from a group of $2n$  (imagine dividing a group of $2n$ into $2$ groups of $n$ people each. I can get $k$ people from group $1$ and another $n-k$ people from group $2$. We do this from $k = 0$ to $n$. Leave Convert to Edit button enabled all the time in the moderator flag list In the moderator flags list, there is a &quot;Convert to Edit&quot; button on answers which are posted by the person who posted the question. This button is disabled at times, and I am not sure that the algorithm used is useful. My proposal is that the Convert to Edit button in the flag list should always be enabled, as it seems fairly frequent that it's disabled when it would be useful. Justification When the button is disabled, it is required that we pop open the answer view in the flag list, and use the regular Mod menu and choose convert to edit there. When we do that, however, we experience this issue:  (that should also be fixed, by the way!) Here are two items currently in the queue:    Note that one has the button enabled, while the other does not. The only differences I can see in those two answers is the age and the number of answers. Neither question has an accepted answer, but the one with the button disabled only has that single answer. I've only just now noticed that difference with these two posts, so I'm not 100% sure if that's the determining factor. 1 It is perhaps coincidental, but these buttons being (dis)abled are reverse of the action needing to be taken. The one with the disabled button needed to be converted to an edit; the one with the enabled button was an attempt to answer their own question. If I'm right about the determining factor here (that the answer is the only one), I'm not sure that's useful. A good bit of the time, an asker will post an answer due to people requesting additional information. In such cases, it seems like it will be likely there are no other answers. 1  Note 1: I was looking to create a userscript to enable these buttons, and I found something that I think is related as to the reason. It looks like the ones which are disabled are on answers which have comments on them. I noticed the title attribute of those buttons says, &quot;There are comments on the post. Please review&quot;. That does seem to make some sense in theory. However, it still seems like I am often converting them, anyway. With the above-linked, separate issue regarding how the 'mod' link on the expanded view works, this is still a bit of an impediment to our workflow. It occurred to me to check if the button was reenabled once you expanded the view, but it does not. Maybe that's an alternate solution. My plan for my userscript is to enable the button, but change the text color (and leave the title attribute), to point out that there are comments. I have parent folder "parent". Inside this folder I have subfolders, and a file named "names.txt". this file includes the names of these subfolders as follow:  Parent_folder folder1 folder2 folder3 folder4 . . . . names.txt   The content of the file "names.txt" is as follow:  folder1 folder2 folder3 folder4 . . .   Inside every folder I have images and I want to apply consecutively 10 scripts on every image ( every script must finish it is job inside every folder, then the second script must be run). These scripts have different names and they are exist in one folder. I set an environment by sourcing a file then I can call these scripts by its name from terminal .At the same time I want to apply this process on all the folders at once. i.e. when script #1 is running, I want it to be running on all the folders at the same time. When it is done and script #2 will start. I want it to start in all the folders at once and so on... In order to achieve this I wrote the following code:  #!/bin/bash path=PATH/TO/THE/PARENT/FOLDER for i in $(cat $path/names.txt); do { script#1 } &amp; { script#2 } &amp; . . .  done   This code is not working efficiently as all the commands are running at once. I want the commands to run on all the folders at once but consecutively.  What I am doing wrong? Are there any significant plot elements in Fantastic Beasts that will be missed if I haven't seen the Harry Potter films? I have read all seven Harry Potter books, and The Cursed Child1, but I've only watched the first two movies.  Do I have to watch all eight movies in order to fully understand the significant plot points found in Fantastic Beasts and Where to Find Them?  Is there anything that I'll miss by not having seen them?    1 Waste of money, that. If I have the following code:  MyClass pClass = new MyClass(); pClass.MyEvent += MyFunction; pClass = null;   Will pClass be garbage collected? Or will it hang around still firing its events whenever they occur? Will I need to do the following in order to allow garbage collection?  MyClass pClass = new MyClass(); pClass.MyEvent += MyFunction; pClass.MyEvent -= MyFunction; pClass = null; Is there a unique solution for this quadratic matrix equation?