convergence of $\sum_{l=2}^\infty{\frac{1}{l^m}}$
Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$
Gerund preceded by a genitive? Is this sentence actually grammatical?     You know your having a rough day when kittens don't even make you smile.   The writer of this sentence may intend to mean you're instead of your but I'm just wondering if having a gerund preceded by a genitive is considered grammatical in this case.  If the sentence above is ungrammatical, why is this sentence grammatical?      He resents your being more popular than he is.   What's the difference between these two sentence that determines one to be grammatical and the other to be not?  This is how I analyze the two sentences and I can't see the difference:   "You know it when kittens don't even make you smile" for "it" = "your having a rough day" "He resents it" for "it" = "your being more popular than he is" What does ;; do in sh? A tank of 425 liters capacity has been filled with water through two pipes, the first pipe having been opened 5 hours longer than the second. If the first pipe were open as long as the second pipe, the first pipe deliver half the amount of water delivered by second pipe; if the two pipes were open simultaneously, the tank would be filled up in 17 hours. How long was the second pipe open?    a. 10 b. 12 c. 15 d. 18 MikTeX 2.9, install packages on the fly? "They're going gorilla watching" - is there a verb at the end of the sentence? In this sentence:     They're going gorilla watching.   what's the function of the verb watching?   Is it a verb? a noun? ... maybe gorilla watching means "mirador de gorilas", in spanish?  In English, I have not seen a verb at the end of a sentece. How to choose a specific weight from a font family using fontspec and XeLaTeX Force NTFS permissions inheritance for newly created files Prettify OS X console by coloring output? OS X terminal is not using colors for highlighting output from different commands, for example ls.   When compared with other systems like Ubuntu, this is a big loss of accessibility/productivity.  What solutions do I have to improve this experience? For the moment I found only this guide for  but I am looking for a more complete solution. Minkowski's complex Euclidean space vs. the real pseudo-Euclidean version Automating "Stay Awake" I use my personal Android smartphone for development. Every morning, I go to Settings \ Developer Options \ Stay Awake and check the box.  At the end of the day, I uncheck the box — since otherwise any notification (including the "fully charged" one) will trigger the screen on, and it will stay on from that point on.  How can these two operations be automated? I am a freelancer programmer and so far I have only worked on projects on Windows platform. I don't have much idea about OS X hardware. I am not looking for a very high end configuration but integrated development environment and one database server should just run fine. I have a Dell laptop i7 processor, 8 GB RAM, 1 TB HDD; I am fairly satisfied with the performance. I am mainly going to use it for app development and not for any rich graphics app development.  What are the minimum and recommended hardware requirements for Xcode development on OS X? Chrome 100% CPU usage, again - software_reporter_tool.exe software_reporter_tool.exe belonging to Chrome running at 100% cpu, fan full speed.  Deleting it doesn't fix anything, because it gets redownloaded again! Disabling "Protect me from bla bla" doesn't make it stop running.  Please anyone have any solutions? How to get glare on "Quick Smoke" (Fire & Smoke explosion) In a case of mine a very high (compared to the video stream) resolution graphical frame (saying "access denied" (when my IP changes)) can appear instead of a video stream, which makes VLC window to resize, getting bigger than my screen is. This annoys me. Can I set up VLC to never resize its window when I don't do it myself and resize content instead? In CodeGear Delphi 2007, how can I turn specific warnings and hints off? I am attempting to turn off H2077 - Value assigned to 'varname' never used. What does $b_i\mid b_{i+1}$ mean in this context? In the computational topology literature, the reduction algorithm for computing the Smith normal form of a boundary matrix uses the notation $b_j &gt; 1 \: \text{ and }\: b_j\mid b_{j+1}$ in the context of the diagonal elements of the Smith matrix. Can anyone give me an idea for what it means? Let $(X, d(x,y))$ be a complete metric space. Prove that if $A\subseteq X$ is a closed set, then $A$ is also complete.  My attempt: I tried to prove that every Cauchy sequence $(b_n)$ of points of $A$ converges to a point $b\in A$. However could not figure out the exit way. Maybe I am on the wrong track. Could you please help me?  edit: More from my attempt:  Suppose $A$ is a closed set and let $(x_n)$ be a sequence of points $A$ such that $\lim_n x_n\to b$.  Suppose now that $A$ has the property that $b\in A$, whenever $x_n$ converges to $b$. We know that every element of $x_n$ which is convergent in $X$ also converges to a point in $A$. Since $x_n$ is a Cauchy sequence in $A$, it must converge to a point $y\in A$. But the limit of a convergent sequence is unique.   Take $x\in A$ and select an appropriate $n$, which enables $x_n$ to converge to a point $x$ in $X$. Since the limit is unique, it must follow that $x=y$. Thus $x\in A$ and $A$ is closed.   If $A$ is a closed subset of $X$, then any Cauchy sequence of a point in $A$ is convergent in $X$ and hence converges to a point in $A$. Thus $A$ is complete. Show that $\tan{(\pi/7)} \tan{(2\pi/7)}\tan{(3\pi/7)}=\sqrt{7}$ How to sail downwind faster than the wind? Recently a  set a record for sailing a wind-powered land vehicle directly down wind, and a speed faster than wind speed. Wikipedia has a  talking about it, but it doesn't explain exactly how it works.  I can imagine some configurations, but I don't get how a system could be set up to push forward with a tailwind (when the cart is moving slower than the wind) and push forward with a head wind (when the cart is moving faster than the wind).  Could someone help explain how this is possible? I've installed reCAPTCHA module but spam still bypass my security and leave nasty comments. What should I do? How to block them?