Limit Laws for a Partial Sum of an Infinite Series
Can't we compute $\lim_{n \to \infty}\frac{1+ \cdots +n}{n^2}=\lim_{n\to\infty}\frac{1}{n^2}+\cdots +\lim_{n\to\infty}\frac{n}{n^2}=0$?
Why I can't edit "Your Communities" list in stackexchange.com and Area 51? I can edit &quot;YOUR COMMUNITIES&quot; in the upper left drop-down menu from any of my communities: (e.g. )    But in  portal there's no &quot;edit&quot; link:    Why is that and how can I edit the list in  portal? I had studied group theory a year ago, but still could not understand the proof involving The Correspondence theorem.  let $G$ be a group and let $N⊴G$, where  $N⊴G$ indicates that $N$ is a normal subgroup of $G$. Then there is a bijection from the set of subgroups $A$ of $G$ that contain $N$ onto the set of subgroups $\overline A $=$A/N$ of $G/N.$  Can anyone explain the steps to me in:  $(1.)$ a very easy-going manner so that I understand importance of each step. $(2.)$I don't require the proof but only how the step follow each other .  I'll be thankful if there is anyone kind enough to do this.... How to hide "internal" polygon boundaries? I have two files in a directory. One has correct permissions and the other has not. Is there a way I can "copy" the set of permissions from one file to another? Is it a good practice to place C++ definitions in header files? My personal style with C++ has always to put class declarations in an include file, and definitions in a .cpp file, very much like stipulated in . Admittedly, part of the reason I like this style probably has to do with all the years I spent coding Modula-2 and Ada, both of which have a similar scheme with specification files and body files. I have a coworker, much more knowledgeable in C++ than I, who is insisting that all C++ declarations should, where possible, include the definitions right there in the header file. He's not saying this is a valid alternate style, or even a slightly better style, but rather this is the new universally-accepted style that everyone is now using for C++. I'm not as limber as I used to be, so I'm not really anxious to scrabble up onto this bandwagon of his until I see a few more people up there with him. So how common is this idiom really? Just to give some structure to the answers: Is it now The Way™, very common, somewhat common, uncommon, or bug-out crazy? Did Dolores Umbridge Have Any Association with Voldemort (or Death Eaters) before His Return? I noticed that Dolores Umbridge was born during the first Wizarding War, so it's very likely she wasn't a Death Eater then (but she is pretty evil -- who knows?).  After that Voldemort was not around in a way that could affect many people, and most wouldn't know he was planning to rise again.  During that time, and up through Voldemort's return (in Goblet of Fire), did Umbridge have any connection with the Death Eaters or with Voldemort?  Was she doing what she did on her own, or was it because of an association with Voldemort or his allies? I would like to format a price in JavaScript. I'd like a function which takes a float as an argument and returns a string formatted like this: &quot;$ 2,500.00&quot;  What's the best way to do this? Difference between $\sum_{i=1}^{k}{\frac{s_i}{kn_i}}$ and $\frac{\sum_{i=1}^{k}{s_i}}{\sum_{i=1}^{k}{n_i}}$ I am getting confused at how to calculate the average probability.  Suppose we repeat a kind of binary survey $k$ times each of which was done on a completely separate sample group.  For each $i^{th}$ group, $i=1,2,3,...,k$, Let $n_i$ be the number of samples and $s_i$ be the number of positive results. With this we know that the probability $p_i$ of the positive result for the $i^{th}$ group is $s_i/n_i$  With my ignorance, I happened to use two ways to calculate the average probability in my work wrongly assuming that they are the same:  $$\frac{p_1+p_2+p_3+...+p_k}{k}$$ and $$\frac{s_1+s_2+s_3+...+s_k}{n_1+n_2+n_3+...+n_k}$$  I don't know which is the correct way to calculate the average probability. So could you please explain the difference of these two and when to use which? On my MacBook Pro I have found that several applications seem to share the value for a find.  For example, I searched 'applestackexchange' in Safari's unified address bar, then I switch to Xcode and click into a Find field.  The first word of my search in Safari then appears there.  iTerm2 and Sublime Text 2 also seem to participate in this sharing.  Running latest Mountain Lion on  a 2012 MBP. Why are 50mm Lenses Cheaper? TSA requirements regarding ID If the name on my boarding pass lists Debbie and my Drivers License lists Debra, will this be a problem? number of prime ideals in algebraic integers How to call gdal_translate from Python code? Is it possible to use some gdal API to call gdal_translate from Python code? I do not mean simply executing the gdal_translate.exe from the file system, but rather call it somehow in code so I do not need to know the exact directory the gdal_translate executable is in? How to pass an array of Uri between Activity using Bundle Ubuntu 20.04 interface font too small after restart, even with high scaling factor So I was configuring VS code on my Notebook when suddenly I noticed that my interface font got really small on my desktop, terminal and files. I use Gnome Tweaks Tool to increase my scaling factor to 1,15 , but, even though it was active, the font stayed tiny. The solution I found was moving my scaling factor, which instantly fixed my font issue, however, every time I restart my Notebook the font gets tiny again and I have to repeat the process. I tried reseting my font configuration using this command, but had no success. gsettings reset org.gnome.desktop.interface text-scaling-factor  I also tried using the command dconf reset -f /  Finally, I tried using dconf-editor, but still had no success altering the font rendering DPI or changing the scaling factor. All of the steps I mentioned fixed my fonts temporarily, but the issue still persists when I restart my Notebook I am out of ideas and even considered getting a fresh Ubuntu install by now... My video card: NVIDIA GeForce GTX 1050 GPU I have been playing with the settings for the text object recently, and every time I add a new text object, I need to change the font to my favorite one. (Eurostile LT ExtendedTwo) Is there any way to change the default font so I do not need to change it every time I add a new text object?   Example: Keying node in compositor. How can I associate the alpha channel so that it works with the Alpha Over Node? (As a follow-up to  question) I'm trying to overlay a a green-screen plate over a 3D background. I create the key with a Keying node and the overlay with an Alpha Over node. I can get a clean key by enabling Convert to Premultiplied.  Whereas if I leave the Convert to Premultiplied off, the key does not work.  The reason for this, as explained on :  The Alpha over node actually assumes associated alpha and performs the correct over operation, which is FG+(1-FG.a)BG.RGB. Unassociated alpha can do none of these things. [...] it is prudent to manually associate your alpha and control the merging to prevent data loss in the RGB channels. I still stand by the advice, however; always try to maintain a consistent state of alpha, and when you deviate, do so under scrutiny.  So the question is: what is the correct way to associate the Alpha Channel crated by the keying node, so that it works with the Alpha over node without resorting to the convert to premultiply option?   green screen element can be downloaded from the mango project Can the 'bin size' in a histogram be thought of as a regularity constraint? How can I change the margins for only part of the text? Sometimes the margins need to be changed for a particular page, paragraph, or other section of text.  For example, if I'm writing a letter and want the left margin to be almost at the right side of the page for the four lines of my address, or if I am combining prose with poetry and want the poems to be indented relative to the rest of the work.  What is the best way to change both left and right margins on the fly? Exact relationship between electric field and intensity I often see $I(t)\propto |E(t)|^2$. What is the exact form ? Which constants are missing to make this an equality and why are they so often omitted ?