Let us suppose we have a triangle with base $j$ to $j-1$ and height of $1/j$ for $j=2,...,n$. Then, $$\frac{1}{j}(j-(j-1))\lt\int_{j-1}^{j}\frac{1}{x}dx$$ Why $\frac{1}{j}(j-(j-1))\lt\int_{j-1}^{j}\frac{1}{x}dx$?
In an Econometrics - Time Series textbook (Hamilton), I saw the claim, without proof: Consider the sequence $ \psi_j $, then, for $ x &gt; j $, $ \frac1j &gt; \frac1x $, and therefore: $$  \frac1j &gt; \int_{j}^{j+1} \frac1x dx $$ for $x$ &gt; $j$. When I plug in numbers for $j$, I see that this is indeed satisfied. But I'd like to know the intuition behind this or a proof.
"Never say die?" What's the deal with this idiom? I know what it means, but cannot wrap my head around its grammar structure. The sentence structure of     "Never say die"   looks like that of     "Never admit defeat"   but it does not make the same grammatical+semantical sense.  Is it an archaic use? Is it a contraction of a longer phrase? Was it originally misspoken by someone of note and survived memetically? What are its roots?  Edit: Well, the comments made me realize that I worded my question poorly. I'm not saying I think this phrase is ungrammatical. I'm asking how the heck "saying 'die'" came to mean what it means. How to completely disable auto-save and versions in Mac OS X Lion? I hate the automatic saving and versioning in Mac OS X Lion.  Is there a way to completely turn it off?  (This has been asked before , but the answer and the linked article are completely wrong, so I thought I'd try again.) I have a few satellites and stations orbiting my planets, but I'd like to add onto them. I've seen numerous videos on using docking clamps to attach my new ship to the existing station, but can I do that on a part-by-part basis? For example, I take an extra fuel tank up there, can I attach just the tank to my station, then send my vessel back to Kerbin? I'm not talking about refueling, but rather transferring individual parts from my ship to the station in orbit.  I'd prefer to not have to use an unmanned probe and jettison everything but the desired part in space after docking if possible, but if that's the only way I guess I don't have a choice. This is, how I download various master branches from GitHub, and I aim to have a prettier script (and maybe more reliable?).  wget -P ~/ https://github.com/user/repository/archive/master.zip unzip ~/master.zip mv ~/*-master ~/dir-name   Can this be shorten to one line somehow, maybe with tar and pipe?  Please address issues of downloading directly to the home directory ~/ and having a certain name for the directory (mv really needed?). How to text-to-speech output using command-line? How to get speech output from entered text by using command-line?  Also facility to change speech rate, pitch, volume etc using simple command. What are "schlieren" in English? The German word for these patterns is "Schliere, pl. Schlieren", and apparently this word has entered the English language as a loanword (cf. article  in Wikipedia or entry  in OED). But the German loanword is probably not something most native speakers of English understand without the help of a dictionary, so what would you call, or how would you describe, the following patterns?  Concrete:    Air:    Oil on water:    Coffee and milk:    Paint:    Smoke:    Rock:    Marbled paper:    And so on.    "Waves" is not applicable, because waves are vertical undulations of the surface, while these are a pattern on, or inside, the surface level, with no amplitude.  "Striæ" does not really fit either, because those refer more to the parallel layers, than to the curly forms. Tabs in output file written by xelatex and pdflatex are different Examples:   In a car, van, etc. On a bus, boat, motorcycle, etc.   How can one decide which preposition to use? Is memorization the only way or is there a better way?  Note: People generally explain this by either distinguishing between open and closed vehicles or between large and small vehicles. However, the examples I've given defeat both explanations. I have two different bookings on British Airways and Emirates.  Booking 1 (British Airways): Washington (IAD) -> London (LHR) -> Dubai  Booking 2 (Emirates): Dubai -> Hyderabad  I have a 5 hour stop in Dubai but do not have a visa to leave the airport. British Airways says they won't through-check my bags. Emirates says it won't be a problem since Emirates and BA are partner airlines.   Not sure what I should do at this point. My flight is tomorrow so its too late to apply for a visa. I have actually done with before without a problem on the same two airlines. The only difference was that it was on a single ticket booked through a travel website. I have installed Ubuntu Server on an old desktop, and I'm new to this.  I had set it up and was trying to connect through Putty but that was not working.  When I tried ifconfig there was no eth0 IP address, and logging onto the router showed the only device connected by ethernet is the printer.  However, the ethernet cable is plugged into the server and is showing the green light by it.  I tried opening /etc/networking/interfaces but this shows the following:  source /etc/network/interfaces.d/*  # The loopback network interface auto lo iface lo inet loopback   I tried adding 'auto eth0' after this and saving but made no difference, and have tried sudo ifconfig eth0 up which didn't do anything either.  sudo lshw -C network says that the network is DISABLED.  Any ideas to help get this up and running?  EDIT: Running command lspci -knn | grep Eth -A3 results in the following:  02:05.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd. RTL-8110SC/8169SC Gigabit Ethernet [10ec:8167] (rev 10)          Subsytem: Gigabyte Technology Co., Ltd GA-MA69G-S3H Motherboard [1458:e000]          Kernel driver in use: r8169          Kernel modules: r8169 I know this is a total noob question but then again, when it comes to PostgreSQL I am a total noob...    I have installed the OpenGeo suite on my computer, running Windows 7, 64-bit.  I have downloaded the tutorial data and have been working through it.  I would like to access the database that I have made with ArcGIS so I can start learning about how that works.  However, when I try to connect to the database using the "Add Database Connection..." I can't seem to figure out what to put in for 'Instance' - everything I've tried (localhost, localhost,54321, my IP address with and without the port, my computer name) doesn't seem to work.  I installed and created the nyc test database according to all the instructions in the OpenGeo tutorial pages but am at a total loss here.  I know this must be an easy thing to deal with, I just can't get my head around it.  Can someone with more experience throw me a bone on this one?    Frustratingly enough, QGIS connects to the database with almost no effort and everything works fine - but my organization is heavily ESRI focused and switching to QGIS is a long shot...  EDIT Thanks everyone for pitching in on this.  I really appreciate the advice, it's what makes this site great.  I have installed    OpenGeo Suite 3.0.1 with no extensions ArcGIS Desktop 10.1, no license for server   I just went to the Opengeo website and downloaded the Windows installer and installed the default configuration. I have tried every permutation for the instance name that I can think of - using colons and commas, my machine name, localhost, postgresql, server, my IP address, random curse words, anything I can think of.  I have been using port 54321 instead of 5432 because that is what the OpenGeo workshop told me to set up my 'nyc' practice database to use.    It seems that localhost should work, here is a shot of the server properties from pgAdmin: Under Windows, most applications and application data are stored in a special directory known as C:\Program Files (and occasionally C:\Program Files (x86)). What is the Ubuntu/Linux equivalent to this path? Is there even one? What is the difference between "why are you worried" and "why you are worried"? What is the difference between these two sentences?     Why are you worried?   Why you are worried?   Which one is correct? remove oldest files I`m trying to delete old files from directory and leave only 3 newest files.   cd /home/user1/test  while [ `ls -lAR | grep ^- | wc -l` &lt; 3 ] ; do      rm `ls -t1 /home/user/test | tail -1`     echo " - - - "  done   something is wrong with conditional statement. Why can one bend glass fibers without breaking it, whereas glasses one comes across in real life is usually solid?  Is there also a good high-school level explanation of this? Formal basis for variable substitution in limits Some cards in magic has something happen for each {something}. For instance:  makes you gain 2 life for each creature you control. Tapping  adds one green mana to your mana pool, and you gain 1 life for each non-land card revealed.  How is this life gained: all at once or "one bit" at a time? Say I tap Selvala and 1 land, and 2 non-lands are revealed: do I gain 2 life, or do I gain 1 life two times? Would  have one or two +1/+1 counters placed on it if I had it in play at that point? Does Windows log programs that have been run/called? I would like to find a way to construct uncountably many non-isomorphic linear (total) orderings of natural numbers (as stated in the title).  I've already constructed two non-isomorphic total orderings. They are $$ 1 &lt; 2 &lt; 3 &lt;\ldots\\ 1 &gt; 2 &gt; 3 &gt;\ldots $$ where "$&lt;$" and "$&gt;$" are  strict total ordering relations.  Thanks in advance for your help. Will I still receive OTA updates after rooting my Nexus 5?