How to find the sum of the following infinite series?  $$\frac1{2 \cdot 3 \cdot 4}+\frac1{4 \cdot 5 \cdot 6}+\frac1{6 \cdot 7 \cdot 8}+\cdots$$  I can write its partial sum but am unable to proceed.
What is the limit of series $\frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{4 \cdot 5 \cdot 6} + \frac{1}{6 \cdot 7 \cdot 8} + \cdots $?  The $n$th summand is $\frac{1}{(2n)(2n + 1)(2n+2)} = \frac{1}{4} \frac{1}{n(2n+1)(n+1)}$.  I have tried expressing this as a telescoping sum, or as the limit of Riemann sums of a partition (the usual methods I normally try when doing this type of question- what are some other strategies?)
What is the fastest algorithm for finding the natural logarithm of a big number? Does Moore's law apply to quantum computing? Plain and simple. Does  apply to quantum computing, or is it similar but with the numbers adjusted (ex. triples every 2 years). Also, if Moore's law doesn't apply, why do qubits change it? As mentioned in many of my previous questions, I'm working through K&amp;R, and am currently into the preprocessor. One of the more interesting things &mdash; something I never knew before from any of my prior attempts to learn C &mdash; is the ## preprocessor operator. According to K&amp;R:     The preprocessor operator ##   provides a way to concatenate actual   arguments during macro expansion. If a   parameter in the replacement text is   adjacent to a ##, the parameter is   replaced by the actual argument, the   ## and surrounding white space are   removed, and the result is re-scanned.   For example, the macro paste   concatenates its two arguments:      #define  paste(front, back)  front ## back      so paste(name, 1) creates the token   name1.   How and why would someone use this in the real world? What are practical examples of its use, and are there gotchas to consider? Easy way to use variables of enum types as string in C? Here's what I am trying to do:    typedef enum { ONE, TWO, THREE } Numbers;   I am trying to write a function that would do a switch case similar to the following:   char num_str[10]; int process_numbers_str(Numbers num) {   switch(num) {     case ONE:     case TWO:     case THREE:     {       strcpy(num_str, num); //some way to get the symbolic constant name in here?     } break;     default:       return 0; //no match   return 1; }   Instead of defining at every case, is there a way to set it using the enum variable like I am trying to do above? The  says that public class SomeObject {   public void SomeOperation()   {     lock(this)     {       //Access instance variables     }   } }  is &quot;a problem if the instance can be accessed publicly&quot;. I'm wondering why? Is it because the lock will be held longer than necessary? Or is there some more insidious reason? What is the closed form for $\sum_{n=1}^\infty \frac1n - \frac1{n+1/p}$? A while ago, I started to look at expressions of the following form: $$ S_p:=\sum_{n=1}^\infty \frac1n - \frac1{n+1/p}, $$  where $p$ is prime, because otherwise things get too complicated for me at the moment.  What I found so far is the following: $$ S_p= p -\log(2p) - \frac12 \pi \cot\left(\frac\pi p\right) + T_p, $$ where $T_p$ contains $\frac{p-1}2$ terms $t_{p,k}$ of the form:  \begin{cases} \pm 2 \sin\left(\frac{k\pi}{2p}\right) \log\left(\sin\left(\frac{k\pi}{2p}\right)\right) \\ \pm 2 \cos\left(\frac{k\pi}{2p}\right) \log\left(\sin\left(\frac{k\pi}{2p}\right)\right)\\ \pm 2 \sin\left(\frac{k\pi}{2p}\right) \log\left(\cos\left(\frac{k\pi}{2p}\right)\right)  \\ \pm 2 \cos\left(\frac{k\pi}{2p}\right) \log\left(\cos\left(\frac{k\pi}{2p}\right)\right)  \end{cases}   An example can be found .     How does the closed form for $\sum_{n=1}^\infty \frac1n - \frac1{a+n}$ look like? (EDIT: where $a=1/p$) How do I use F3 to get my coordinates when my laptop makes F3 a special key? I can't get the coordinates on my computer because it keeps changing the brightness.  I have a windows 8 Samsung, and I'm playing Minecraft version 1.6.4. I'm married have a job with stable income.I have 2 kids. I have applied for UK visit visa about 7 years ago and was refused. I can't remember what exactly was the reason for refusal.  I want to travel for tourism to spend 9 days and my budget is around 1300GBP for the trip. I would be spending part of my annual leave in the UK. I have the following people in UK: Mother, Dad, Sister(she's married), Brother (married), Uncle and Aunt.  I'm travelling alone since my wife is pregnant. I haven't travelled since I was in school then.  I will be on my yearly 21 days annual leave during which I will be spending 9days in UK. I can't think of anything more about my circumstance.  So what should a cover letter look like?  secondly I have just 1 account where my salary is paid in monthly and my savings.I will be submitting the 6 months bank statement for the account.  Additional question:  In the income &amp; expenditure where one is asked to indicate if I have income and savings,should I click Yes or No. And if Yes should state the ending balance as at the time of my application. Should I follow what was mentioned in the “duration of stay” or in the "validity of the visa"? I'm from Tunisia and I got a Schengen visa from the Austrian Embassy for  28 days, the “duration of stay”, and the number of entries is 1. But the validity of the visa is from 02/08/2015 to 13/09/2015.  Do I have to stay only 28 days, for example from 02/08/2015 to 29/08/2015 or can I stay there until 12/09/2015 ?  And if I have got to stay only 28 days, then can I choose the days of stay, that's mean can I stay in Austria for example from 09/08/2015 to 05/09/2015? How to cluster time series? I have a question about cluster analysis. There are 3000 companies, which have to be clustered according to their power usage over 5 years. Each company has values for every hour during 5 years. I would like to find out if some companies have the same pattern in usage power over the time period. The results should be used for daily prediction of power usage. If you have some ideas how to cluster time series in SPSS, please share with me. If I build a package from source how can I uninstall or remove completely? I used source code to build one package such as below:  ./configure --prefix=/usr --sysconfdir=/etc --localstatedir=/var --libexecdir=/usr/lib --with-package-name="Myplugin" --with-package-origin="http://www.ubuntu.org/" --enable-gtk-doc --disable-static make make install   But unfortunately, i discovered that its the latest version, and has lot of bugs, so i need to remove it/uninstall it. But how can i do so? I tried make clean; make uninstall but still i see it exist:  # pkg-config --list-all | grep Myplugin myplugin-.... $ ls /usr/lib/myplugin/libXYZ.so exist....   How do you remove this now? How do I change folder timestamps recursively to the newest file? Is the reason why the time evolution operator is unitary based on purely physical arguments, i.e. that the physical processes that an isolated system undergoes shouldn't depend on any particular instant in time (homogeneity of time); thus two experimenters who conduct the same experiment starting from the same initial state, but at different times, should have the same probability amplitude for that state?! Or is there some mathematical argument as well?  Also, is the reason why the time evolution operator is linear implied by the superposition principle (as an arbitrary state can be expressed as a linear combination of basis states the operator should act linearly as otherwise the state as a whole would evolve differently to the superposition of states that it was initially represented by)?! I need to print an image at a certain size. What dimensions and resolution should I use? (This question is meant to be a resource to direct others to given that this is a common question.)  Given the task of creating an image that will be printed at a certain size, what pixel dimensions and resolution should I be setting my file at? Scanning Java annotations at runtime What is the best way of searching the whole classpath for an annotated class?  I'm doing a library and I want to allow the users to annotate their classes, so when the Web application starts I need to scan the whole classpath for certain annotation.  Do you know a library or a Java facility to do this?  Edit: I'm thinking about something like the new functionality for Java EE 5 Web Services or EJB's. You annotate your class with @WebService or @EJB and the system finds these classes while loading so they are accessible remotely. Stack Overflow, Stack Exchange, Area 51: What's the difference in a nutshell? Stack Overflow, Stack Exchange, Area 51: What's the difference in a nutshell? I've read the faq's and they all seem to be the same to me - Just Q &amp; A on a variety of topics.  What's the difference in a nutshell? If I have a software engineering question, which site is best for me to use? How to download Bootcamp drivers without Bootcamp assistant? I'm setting up my MacBook Air for a Windows-only boot setup. This was easy enough. It already had rEFIt on it, I just connected a Windows 7 boot thumb drive, and away I went.  However, it seems Apple does not let you download Bootcamp drivers outside of their assistant tool. Reinstalling OS X just for the drivers is ridiculous.  This laptop no longer has OS X on it at all. How do I get the required drivers? I cannot find them anywhere on Apple's support pages.  Every other manufacturer has their drivers available for download.   How do I download the Windows drivers for my 2010 MacBook Air without the BootCamp Assistant? BasicTeX - how to identify packages that need to be added after an upgrade I'm using the BasicTeX distribution on OS X and recently upgrade from the 2011 to the 2012 distribution. This forces me to add the packages that I had added over time to the BasixTeX 2011 installation by pulling them from the online repository with tlmgr. Is there a good way to list these packages that are not part of BasicTeX but are part of the complete TeXLive? ggmap: create circle symbol where radius represents distance (miles or km) We were asked to prove the following:  Let $ A $ be an $n \times n$ matrix with $n$ distinct real eigenvalues. If $AB=BA$, show that $B$ is diagonalizable.   It was suggested I show that an eigenvector of $A$ is also an eigenvector of $B$. I am both having trouble doing this and failing to see how I would complete the proof after. Any help would be appreciated.  Thanks