How do I find de sum of this serie using the partial sums?  $$\sum_{n=1}^\infty \log\left[\frac{(n+1)^2}{n(n+2)}\right]$$  Thanks!
I am having trouble with (a) and (b):  I evaluated (a) and I got that sun of the first three terms does not equal $\ln(5/8)$. I got that it is equal to $\ln(3/4) + \ln(2/3) + \ln(15/16)$ which is not equal to $\ln(5/8)$.  I am also having trouble with part (b). I cannot figure out how to calculate the math to cancel out the $(k+1)^2$ and turn it into $2(n  +1)$.  Any help would be amazing.
Quantum mechanics scattering theory Gaussian Integers and Quotient Rings Show that $\mathbb{Z}[i]/(2+3i)$ is a finite field.    By applying some isomorphisms and realizing $\mathbb{Z}[i]$ as the quotient ring of $\mathbb{Z}[x]/(x^{2}+1)$, I found that:  $\mathbb{Z}[i]/(2+3i) \cong \mathbb{Z}[x]/(x^2+1,2+3x)$  However, I run into trouble as I can't divide $x^{2}+1$ by $2+3x$ in $\mathbb{Z}[x]$. What would be another way to approach the problem?  Thanks for the help. How can I recover data from a clicking/dying external drive? Windows asks me to reformat every time I plug it into my computer, so I fired up Ubuntu via VirtualBox (selecting the external hard drive in the USB settings for VirtualBox beforehand) to see if I can access it there.  Does anyone have any suggestions for me? I have over 1TB of data on this drive and do not want to lose it. When I plug it into an electrical outlet, the light turns on, and when I plug it into my laptop, I can feel it clicking. How can I create a Windows bootable USB stick using Ubuntu? I'd like to create a Windows bootable USB stick, but I don't have a Windows machine with me at the moment to do so. How could I achieve the same using Ubuntu? How to remove zsh alias? Reading the book on which I'm learning English, I've come across a weird (for me) omitting the "to" in sentence: "I'd better go meet with him then."  Say, if it were "I'd better go to meet with him then.", would there be any difference between the two sentences? When I should prefer using one instead the other?  P.S. Sorry for my bad English. If I made a mistake somewhere, please correct me. Why can a discontinuous function not be differentiable? I don't really understand why a discontinuous function cannot be differentiable.  In Stewart's Calculus, the definition of a function $f$ being differentiable at $a$ is that $f'(a)$ exists. Earlier it gives the definition of the derivative as $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. It also has a theorem that if $f$ is differentiable at $a$, then $f$ is continuous at $a$. There is also a definition that a functions $f$ is continuous at a number $a$ if $\lim_{x\to a}f(x)=f(a)$.  Take for example the very simple function:  $$ f(x)=\begin{cases} x+1 &amp; x\geq0, \\ x &amp; x&lt;0.  \end{cases} $$  It is discontinuous at $x=0$ (the limit $\lim_{x\to 0}f(x)$ does not exist and so does not equal $f(0)$), but if I find the derivative using the limit above, I get the left and right limits to equal $1$. So therefore, the derivative exists.  According to the book, the function shouldn't be differentiable at $x=0$ as it has a discontinuity (continuity is a necessary condition of differentiability). What am I doing/understanding wrong? Installing grub2 in efi-mode in usb flash drive I've two different OS, which one is ubuntu. I'm using a PC, NOT a Mac. I use Chimera bootloader that don't find Ubuntu. So, i'd like to install grub 2 in EFI-mode on flash-drive USB. In this way, i can load Ubuntu only if the usb flash-drive is connected.  I followed this guide  replacing "/dev/sda" in  sudo efibootmgr --create --gpt --disk /dev/sda --part 1 --write-signature --label "GRUB2" --loader "\\EFI\\grub\\grub.efi"   with the usb-pen directory. Then, i've copied the grub.cfg file that exists in Ubuntu partition.Result? Grub2 loads, but when i select the menu entries, grub2 doesn't find any command, or disks. How can i fix this problem?  P.S. Here, there are the photo of the error(when i select ubuntu entry in grub2), the usb-drive's contents and the grub.cfg file Is it normal for a fast prime to exhibit purple fringing in middle of the image? I have just bought a Nikon 50mm f/1.8D prime lens.  I took this picture (leg of my glasses frame on a mouse, and a LED desk lamp is on top of it) : f1.8, 1/80s, ISO 200   The purple fringing is right in the middle of the picture. Does it happen very often to this lens? Lebesgue density theorem in the line MySQL cartesian product between two SELECT statements I used to use a lot ctrl+alt+numpad to move windows, how can I use this feature on ubuntu 18 like it was default before?  Thank you in advance Square version of \cdot (small black square)? I'm looking for the symbol that's similar to \cdot in math mode, but instead of a small circle, it's a square. Like this: Using simple quotes "" instead of \text{} Is there a way to use newcommand or some other nice technique to transform any occurence of  "bla"   in math mode into  \text{bla}   The reason I ask is because it is easy, and doesn't interrupt the flow of reading the (La)TeX code as much.  I would like to keep Xe(La)TeX's nice quote behavior (ie it can use the regular "" instead of backticks to quote some text outside math mode). What item bonuses apply to my follower? There are a lot of different side effects an item can have, and I assume that things like magic find have no influence on any of "our" found items at all.  However, I'm not sure about other things. For example:   Do followers get life-steal on weapons, too? Does increased orb healing make any difference? What about +experience after kills? Will my follower level up faster?   and probably a lot more.  Is there some general rule as to what effects of an item apply to followers, and which don't? C++ templates that accept only certain types Dynamic Symlinks I have built my app for different architectures and would like to create a "dynamic symlink" which takes me to the right version based on a variable.   If the machine I am currently logged in to, is x86, then the symlink should take me to that build.  A mocked example of what I want to do:  $ uname -i x86_64  $ ls -ltr  lrwxr-xr-x  1 dogbane gg     4 Feb 16 15:40 mylink -&gt; @sys/ drwxrwxr-x  2 dogbane gg  2048 Feb 16 15:40 x86_64/ drwxrwxr-x  2 dogbane gg  2048 Feb 16 15:40 i386/  $ cd mylink  $ pwd -P ~/x86_64   Is this possible?   I know I can use cd $(uname -i), but want something simpler. iPhone asks for Mac password (or Mac asks for iOS password) During iPhone update to iOS 10.3.1, I was asked for my password using the correct machine name for my Mac mini. To be clear, the request was not for my iCloud password, my actual local user account password for macOS. This can also happen in the other direction - setting up a Mac may ask for verification for you to enter your iPhone / iPad passcode (and not your iCloud password)  I could not find any information about this kind of access. I was thinking it was security for iCloud and possibly related to two step authorization or two factor authorization. This is the first time I can think of that I entered a desktop password outside of that actual desktop.  I don't have a screen shot but I recall an implication that some files in my Mac are encrypted and need my password. '... need to enter the password for .'   Are there settings I can I control this assuming it's a legitimate part of iCloud setup process or iOS setup process? (How) Can SE sites change their reputation payout? A recent question on  reminded me that there is a  about awarding 10 rep per question upvote. This is mainly due to the relative difficulty and usefulness of asking/answering being different on the site (or appearing to be, at least).  The highly voted answer on PPCG is "Yes, we should do that," but it's been well over three years since it was posted. The answer there said that (at least at that time) meta.so had this implemented already, so it wouldn't/shouldn't be hard to do.  Given that it's been quite some time, I don't know if that's (still) true. I know that meat has undergone changes since then, but as long as I've been actively participating, meta.se (and meta.so before the split) has given 5 points per question upvote, not 10.  My question is: What would the process to change it look like? It's been discussed on meta.ppcg and agreed upon, so what's the next step? Or is it not feasible at all? Yesterday I started the upgrade from 12.10 to 13.04, but I interrupted it because it was taking too long. Now if I restart it, it doesn't proceed.  Is there any possibility to clean the mess it left when I interrupted it and make a clean restart of this update?