I first expanded the equation into $\frac{1}{(z-i)(z+i)}$, then I apply the theorem that $$\underset{z=i}{Res}\frac{1}{(z-i)(z+i)}=\lim_{z\rightarrow i}(z-i)\frac{1}{(z-i)(z+i)}=\frac{1}{2i} $$ but the correct answer should be $\frac{-i}{2}$, which step is wrong?
Is there an easy way to calculate the residue at one of the poles of a rational expression of the form $\frac{1}{1+z^n}$?  I end up having to add a bunch of polar-form complex numbers in the denominator which I have no idea how to do except to convert them to rectangular coordinates, which becomes really messy.  I feel like there ought to be simple pattern/rule for such a canonical expression.
I have a Toshiba satellite A-200 laptop with a Vista OS on it with 4 NTFS partitions  (C:) Vista  (D:) Entertainment  (E:) Work  (F:) Sources   and I wanted to start using Ubuntu instead. So I tried it first from the live CD and everything was OK and all the partitions were shown and working and so I decided to install Ubuntu to replace Vista on the (C:) drive. After I did that I can no longer find my folders and files on the (D:), (E:), (F:) partitions and the only file system that is shown is one 198 GB although my HDD is 320 GB.  I can't access the lost data on the remaining 120 GB which I hope is still there and not totally lost  I am now working from the live CD but I am unable to install testdisk. Can I recover the Vista partitions by the product recovery CD to get my laptop back to the factory settings? Can I recover the NTFS partitions using a recovery program for Windows or will that make the problem worse?   I need these data badly as I don't have a backup for them. Do I need a visa to transit (or layover) in the Schengen area? How to convert .bib generated by Mendeley for use with amsrefs I'm using TeX live 2010.  I have a .bib file generated from Mendeley, and  I want to use that with amsrefs, but it's confusing and the amsrefs documentation is somewhat hard to follow: how do I massage the .bib file into something that amsrefs can stomach. (I'm using amsart as the document class). Can a service written in .NET self-terminate? I have a service application written in C# and under certain circumstances, I would like it to terminate itself.  This would happen after the service has been running for a while, so this would not be happening in the OnStart() event.  Everything that I have read so far suggests that the only safe way to terminate a service is through the Service Control Manager.  My service runs as Local Service and does not have the rights to start or stop services, so I can't access the SCM from the service itself.  Is there another way to self-terminate while still playing by the rules of the SCM? Software for measuring internet traffic? I'm interested in finding a free piece of software for Windows XP &amp; 7 that allows us to measure both incoming and outgoing internet traffic, but not traffic between users of the local network.  I live in a shared household with three others, and we're interested to see which of us is using the largest amount of our monthly internet quota. We're all happy to install the necessary software.  Any suggestions? The question is:     Show that there are an infinite number of pairs $(m,n): m, n \in \mathbb{Z}^{+}$, such that: $$\frac{m+1}{n}+\frac{n+1}{m} \in \mathbb{Z}^{+}$$   I started off approaching this problem by examining the fact that whenever the expression was a positive integer, the following must be true:  $$\frac{m+1}{n}+\frac{n+1}{m} - \left\lfloor\frac{m+1}{n}+\frac{n+1}{m}\right\rfloor = 0$$  However, I was unable to do much more with this expression, so I abandoned it and started the problem again from a different angle.  Next I re-arranged the expression to state that:  $$\frac{m^2+m+n^2+n}{mn} \in \mathbb{Z}^{+} \implies \frac{m(m+1) + n(n+1)}{mn} \in \mathbb{Z}^{+}$$  And therefore:  $$mn \mid (m(m+1) + n(n+1))$$  However, I'm unsure how to demonstrate there are infinitely many occurances of $(m, n)$ for which this is true. So I'd appreciate any help.   Thanks in advance Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$ Here is an exercise, on analysis which i am stuck.    How do I prove that if $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$, then the sequence $\{F_{n}(x)\}$ is boundedly convergent on $\mathbb{R}$? I am trying to really understand why the gradient of a function gives the direction of steepest ascent intuitively.  Assuming that the function is differentiable at the point in question, a) I had a look at a few resources online and also looked at this   , a popular question on this stackexchange site. The accepted answer basically says that we multiply the gradient with an arbitrary vector and then say that the product is maximum when the vector points in the same direction as the gradient? This to me really does not answer the question, but it has 31 upvotes so can someone please point out what I am obviously missing?  b) Does the gradient of a function tell us a way to reach the maxima or minima? if yes, then how and which one - maxima or minima or both? Edit: I read the gradient descent algorithm and that answers this part of my question.  c) Since gradient is a feature of the function at some particular point - am I right in assuming that it can only point to the local maxima or minima? (In, On or At) GitHub? What's the difference between say:     "The project will be on GitHub",      "The project will be in GitHub" and      "The project will be at GitHub"? How can I tell if two Minecraft mods are compatible? If I want to use two mods together, e.g.  and  (which don't state compatibility with one another), how can I tell if they will work together (and not break Minecraft - forcing me to re-download a fresh copy)? As I understand it, electromagnetic waves have two components which are the result of each other, i.e., when a moving electric charge creates a changing magnetic field at point X then a changing electric field is created at point Y and this repeating process is what creates EM waves, so therefore, it requires no medium. Is my understanding correct?  One thing that I'm surprised to know is that light is also called an electromagnetic wave.  Does this include light of any kind, for example: light from a bulb, a tube, and also from the Sun? How do they contain electric and magnetic fields? How to disable Windows 10 memory compression? my former wrong assumption: it drains my notebook battery and I've no shortage of RAM. In Task manager you see ntoskrnl.exe is eating more CPU time than any other process. this feature does not appear to be optional and came 2 days ago with the 1st Win10 Update (Win10Pro in my case).  Edit:  I want to disable it, at least to exclude it as a culprit for my (unproven) assumptions. That process never came to my attention in the last 2 months and in task manager's long name it's called "system memory and compressed memory". Storing third-party libraries in source control Should libraries that the application relies on be stored in source control? One part of me says it should and another part say's no. It feels wrong to add a 20mb library that dwarfs the entire app just because you rely on a couple of functions from it (albeit rather heavily). Should you just store the jar/dll or maybe even the distributed zip/tar of the project?  What do other people do? C# Equivalent of SQL Server DataTypes For the following SQL Server datatypes, what would be the corresponding datatype in C#?  Exact Numerics  bigint numeric bit smallint decimal smallmoney int tinyint money     Approximate Numerics  float real     Date and Time  date datetimeoffset datetime2 smalldatetime datetime time     Character Strings  char varchar text     Unicode Character Strings  nchar nvarchar ntext     Binary Strings  binary varbinary image     Other Data Types  cursor timestamp hierarchyid uniqueidentifier sql_variant xml table   (source: ) Why are we using a biased and misleading standard deviation formula for $\sigma$ of a normal distribution? I have a file with a number of lines in a file filename.  I want to count how many lines start with character 'a', with 'b' and so on in one go.  What command i should execute.? Heathrow change in terminal flight UK Border control check? I am traveling from NYC to Mumbai via London. Assume my connecting flight belongs to different terminal. If I have to change terminal at Heathrow, do I need to go through UK Border control ? How should a user's death be handled? It is a bit of a morbid concept, but having recently had a friend die unexpectedly and having to deal with the online repercussions of this event has gotten me to thinking about what should happen to my accounts when I kick the bucket.  I haven't had any great insights yet, but here are some things I think are worth thinking about:   an entry in the FAQ telling the person's loved ones what to do with respect to the account would be extremely helpful, specifically who to contact, how to contact them, and what proof is necessary some form of optional date of death on the user account screen some way to mark the user as no longer being able to generate content and blocking any attempt to login as the user.   The danger I see in implementing such features is the possibility of someone faking another person's death and thereby locking a user out of his or her account or worrying his or her friends. How is voltage always able to increase, if electrons move at the speed of light? Biblatex with Biber: Configuring my editor to avoid undefined citations