I am not sure how to approach this question. I tried to move the limit into the sum, but that turns the result to $0$, which is incorrect.
What is the sum of the 'second half' of the harmonic series?  $$\lim_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?$$  More precisely, what is the limit of the above sequence of partial sums?
See these three pages:          All three of them say:     Earned at least X total score for at least 20 answers in the  tag.   This same text also appears on the tooltip for the tag badge:   Also note the double colon after 'gold badge'  As well as on these pages:          But the  clearly states:        Bronze Badge:  You must have a total score of 100 in at least 20 non-community wiki answers to achieve this badge.   Silver Badge: You must have a total score of 400 in at least 80 non-community wiki answers to achieve this badge.   Gold Badge: You must have a total score of 1000 in at least 200 non-community wiki answers to achieve this badge.        Edit:  It has been noted in the comments that other users see different numbers in these pages, but in all cases it appears they see the same number of answers are required for all tag badge description, regardless of the badge's type. As I understand this, firewalls (assuming default settings) deny all incoming traffic that has no prior corresponding outgoing traffic.  Based on  and , reverse SSH tunneling can be used to get around pesky firewall restrictions.  I would like to execute shell commands on a remote machine. The remote machine has its own firewall and is behind an additional firewall (router). It has an IP address like 192.168.1.126 (or something similar). I am not behind a firewall and I know the remote machine's IP address as seen from the Internet (not the 192.168.1.126 address). Additionally, I can ask someone to execute ssh (something) as root on the remote machine first.  Could anyone explain me, step by step, how reverse SSH tunneling works to get around the firewalls (local and remote machines' firewalls and the additional firewall between them)?  What is the role of the switches (-R, -f, -L, -N)? I had an little issue when loggin in with Safari on Mac OS X (El Capitan).   When entering the password in Safari, there is a key symbol that lets you choose among any password that is stored in your keychain. This button interferes with the "forgot?" button that Stack Exchange adds and you can't press the Safari button.     Please move this "forgot?" button below the text field, like most other sites implement this. Here is the .  Picture:   It's nothing special but I cant figure out why I get problems rigging this mesh. It would be important and a great Help! Thanks for reading and I know there are a lot of guides answering this question but I tried all and it doesn't seem to work. I have a SQL query that I am trying to optimize:  DECLARE @Id UNIQUEIDENTIFIER = 'cec094e5-b312-4b13-997a-c91a8c662962'  SELECT    Id,   MIN(SomeTimestamp),   MAX(SomeInt) FROM dbo.MyTable WHERE Id = @Id   AND SomeBit = 1 GROUP BY Id   MyTable has two indexes:  CREATE NONCLUSTERED INDEX IX_MyTable_SomeTimestamp_Includes ON dbo.MyTable (SomeTimestamp ASC) INCLUDE(Id, SomeInt)  CREATE NONCLUSTERED INDEX IX_MyTable_Id_SomeBit_Includes ON dbo.MyTable (Id, SomeBit) INCLUDE (TotallyUnrelatedTimestamp)   When I execute the query exactly as written above, SQL Server scans the first index, resulting in 189,703 logical reads and a 2-3 second duration.   When I inline the @Id variable and execute the query again, SQL Server seeks the second index, resulting in only 104 logical reads and a 0.001 second duration (basically instant).  I need the variable, but I want SQL to use the good plan. As a temporary solution I put an index hint on the query, and the query is basically instant. However, I try to stay away from index hints when possible. I usually assume that if the query optimizer is unable to do its job, then there is something I can do (or stop doing) to help it without explicitly telling it what to do.  So, why does SQL Server come up with a better plan when I inline the variable? Redefining "cited on" string (and others) in biblatex I'm trying to modify the strings biblatex uses to set the "cited on" string used in the bibliography to identify the pages where a given references has been cited.    What I gather from the biblatex documentation (e.g., § 4.9.2.16) is that something like   \renewbibmacro{backrefpage}{&lt;newtext&gt;} \renewbibmacro{backrefpages}{&lt;newtext&gt;}   ought to work, but they have no effect for me (I'd also like to get this working for the "visited on" string, urlseen).  My MWE is unchanged from , with the addition of these (and related lines) just about anywhere. JSF 2.0 File upload I am trying to install Linux headers for Kali Linux on my machine and I have tried every possible solution on the internet but it always show "Unable to locate packages "  root@kali:/usr/sbin# apt-get install linux-headers-4.6.0-kali1-amd64 Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package linux-headers-4.6.0-kali1-amd64 E: Couldn't find any package by glob 'linux-headers-4.6.0-kali1-amd64' E: Couldn't find any package by regex 'linux-headers-4.6.0-kali1-amd64'   Here is my sources.list file :  # Regular Repositories deb http://http.kali.org/kali sana main non-free contrib deb http://security.kali.org/kali-security sana/updates main contrib non-free # Source repositories deb-src http://http.kali.org/kali sana main non-free contrib deb-src http://security.kali.org/kali-security sana/updates main contrib non-free   Uname -a output :   root@kali:/usr/sbin# uname -a Linux kali 4.6.0-kali1-amd64 #1 SMP Debian 4.6.4-1kali1 (2016-07-21) x86_64 GNU/Linux Show that $3p^2=q^2$ implies $3|p$ and $3|q$ This is a problem from "" (specifically, exercise set 2 #9), which is one of my math texts. Please note that this isn't homework, but I would still appreciate hints rather than a complete answer.  The problem reads as follows:     If 3p2 = q2, where $p,q \in \mathbb{Z}$, show that 3 is a common divisor of p and q.   I am able to show that 3 divides q, simply by rearranging for p2 and showing that  $$p^2 \in \mathbb{Z} \Rightarrow q^2/3 \in \mathbb{Z} \Rightarrow 3|q$$  However, I'm not sure how to show that 3 divides p.    Edit:  Moron left a comment below in which I was prompted to apply the solution to this question as a proof of $\sqrt{3}$'s irrationality. Here's what I came up with...  [incorrect solution...]  ...is this correct?  Edit:  The correct solution is provided in the comments below by Bill Dubuque. "POSSIBLE BREAK-IN ATTEMPT!" in /var/log/secure — what does this mean? I've got a CentOS 5.x box running on a VPS platform. My VPS host misinterpreted a support inquiry I had about connectivity and effectively flushed some iptables rules.  This resulted in ssh listening on the standard port and acknowledging port connectivity tests. Annoying.   The good news is that I require SSH Authorized keys.  As far as I can tell, I don't think there was any successful breach.  I'm still very concerned about what I'm seeing in /var/log/secure though:     Apr 10 06:39:27 echo sshd[22297]: reverse mapping checking getaddrinfo for 222-237-78-139.tongkni.co.kr failed - POSSIBLE BREAK-IN ATTEMPT! Apr 10 13:39:27 echo sshd[22298]: Received disconnect from 222.237.78.139: 11: Bye Bye Apr 10 06:39:31 echo sshd[22324]: Invalid user edu1 from 222.237.78.139 Apr 10 06:39:31 echo sshd[22324]: reverse mapping checking getaddrinfo for 222-237-78-139.tongkni.co.kr failed - POSSIBLE BREAK-IN ATTEMPT! Apr 10 13:39:31 echo sshd[22330]: input_userauth_request: invalid user edu1 Apr 10 13:39:31 echo sshd[22330]: Received disconnect from 222.237.78.139: 11: Bye Bye Apr 10 06:39:35 echo sshd[22336]: Invalid user test1 from 222.237.78.139 Apr 10 06:39:35 echo sshd[22336]: reverse mapping checking getaddrinfo for 222-237-78-139.tongkni.co.kr failed - POSSIBLE BREAK-IN ATTEMPT! Apr 10 13:39:35 echo sshd[22338]: input_userauth_request: invalid user test1 Apr 10 13:39:35 echo sshd[22338]: Received disconnect from 222.237.78.139: 11: Bye Bye Apr 10 06:39:39 echo sshd[22377]: Invalid user test from 222.237.78.139 Apr 10 06:39:39 echo sshd[22377]: reverse mapping checking getaddrinfo for 222-237-78-139.tongkni.co.kr failed - POSSIBLE BREAK-IN ATTEMPT! Apr 10 13:39:39 echo sshd[22378]: input_userauth_request: invalid user test Apr 10 13:39:39 echo sshd[22378]: Received disconnect from 222.237.78.139: 11: Bye Bye     What exactly does "POSSIBLE BREAK-IN ATTEMPT" mean?  That it was successful? Or that it didn't like the IP the request was coming from? Is $f(x)=10$ a periodic function? I am not getting satisficatory explanation for this. Clearly $f(x+T) = f(x)$ for all values of $T$.  If we assume it is periodic, does this mean period =  $0$? Why are map tiles being repeated? Using MapServer and Openlayers Do I need a transit visa for a layover in the United Arab Emirates? Vanishing of Taylor series coefficient In a Silverlight application I'm trying to find out when a property on a usercontrol has changed. I'm interested in one particular DependencyProperty, but unfortunately the control itself doesn't implement INotifyPropertyChanged.  Is there any other way of determining if the value has changed? I have an Optimus Elite, and it has a minimum brightness setting. In certain apps, I can reduce the brightness below the safe minimum, but when I leave the app, the brightness is increased to the system-wide minimum.  Using , I can permanently set the brightness below the safe amount, and it will stay even after going to other apps. Or using , I can accomplish almost the same thing even without needing root (it displays a translucent 'filter' over the whole screen, reducing brightness/contrast but not actually changing the backlighting power amount).  What I want to know is where the minimum screen brightness is stored. I have a rooted phone with CyanogenMod 9, so I have permissions to edit anything and could change a config or sqlite entry, but I don't know where this value is stored. What I would like to do is use the status bar brightness slider to be able to slide to the actual minimum of the screen, rather than the OS-controlled minimum. Having an app is useful, but having the functionality without the app seems to be within reach, if only I knew where the OS is getting it's safe minimum setting.  Where is this "safe minimum brightness" setting stored? If I can change it, I can remove my apps and use native controls. Can't understand a proof: Let $a,b,c$ be integers. If $a$ and $b$ divide $c$, then $lcm(a,b)$ also divides $c$  Proof I've been shown:  Since $c$ is a common multiple of $a,b$, it means that $lcm(a,b) \leq c$.  So: $c=q\cdot lcm(a,b)+r$, $q \in \mathbb Z$ and $0\leq r \leq lcm(a,b)$  Since $c$ and $lcm(a,b)$ are common multiples of $a,b$, then so is $r$. But $m$ is the least common divisor, then it means $r=0$.   [Proof taken from: http://www.math.utah.edu/~wood/4400/hw1sol.pdf]  I cannot understand the last step... Why does it imply that $r=0$?   Thank you! How can I enable shell-escape? I need to enable shell escape for a package that I want to use. What is shell escape exactly, and how do I enable it in my editor or compiling toolchain? Links to answers for different editors/toolchains, in alphabetical order:   AUCTeX (Emacs) ,         Note: this is a FAQ post. Related Meta question: . Please add an answer for your own editor/toolchain if it is missing. Thank you! When adding an answer, please indicate if the setting is permanent or per-document, and/or describe how you can add a button, shortcut, menu item, etc. to make it easy to switch between shell-escape and regular compilation. I'm trying to run multiple Nodejs applications on Nginx server running on CentOS 7. I noticed that when I run a Nodejs app on some ports I get an 502 Bad Gateway error in the browser so I checked the error logs:  [notice] 12806#0: signal process started [crit] 12807#0: *13 connect() to 127.0.0.1:7777 failed (13: Permission denied) while connecting to upstream, client: **.**.99.58, server: myapp.com, request: "GET / HTTP/1.1", upstream: "http://127.0.0.1:7777/", host: "myapp.com" [crit] 12807#0: *13 connect() to [::1]:7777 failed (13: Permission denied) while connecting to upstream, client: **.**.99.58, server: myapp.com, request: "GET / HTTP/1.1", upstream: "http://[::1]:7777/", host: "myapp.com"   when I change the app to listen to 8008 for example everything is working fine. I checked permissions and if the process is running as root and everything seems ok. I played with the timeouts as well but no result. Can anyone help? Could it be that Goldbach conjecture is undecidable?