$$\lim_{n\rightarrow\infty}\left(\frac{a^{1/n}+b^{1/n}}{2}\right)^n$$  No idea how to do it please help step by step
Let $a$ and $b$ be positive reals. Show that $$\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$$
Downloading from the main Ubuntu archive is slow even when it's not release day, how can I get apt-get to automatically use a mirror that is close to me? Inheritance of static members in PHP In PHP, if a static attribute is defined in the parent class, it cannot be overridden in a child class. But I'm wondering if there's any way around this.   I'm trying to write a wrapper for someone else's (somewhat clunky) function. The function in question can be applied to lots of different data types but requires different flags and options for each. But 99% of the time, a default for each type would suffice.  It would be nice if this could be done with inheritance, without having to write new functions each time. For example:  class Foo {     public static $default = 'DEFAULT';      public static function doSomething ($param = FALSE ) {         $param = ($param === FALSE) ? self::$default : $param;         return $param;     } }  class Bar extends Foo {     public static $default = 'NEW DEFAULT FOR CHILD CLASS'; }  echo Foo::doSomething() . "\n";  // echoes 'DEFAULT'  echo Bar::doSomething() . "\n";  // echoes 'DEFAULT' not 'NEW DEFAULT FOR CHILD CLASS'  // because it references $default in the parent class :( I am running a web site design service for local small business owner. I want avoid legal issues on the images I use. Usually, the images found on Flickr are forbidden for commercial use. Where would be a normal channel to get pictures for commercial use with minimal cost? How can I validate that the picture I used has no legal risk? Is there a database of Linux compatible hardware? I want to know if there is a centralised community site where all Hardware is listed in terms of compatability with Linux. Something like AppDB for Wine. Is there a word for somebody who cares for his/her child too much? I have just finished watching the 3rd season of .  I am wonder whether there is a word for somebody who cares for his/her son/daughter/child too much, like Dr. Cal Lightman? To ensure that a new chapter starts on a right-hand page, the book class (and the report class with the options twoside and openright) will insert blank pages between chapters if necessary. However, these pages still contain the headers and footers as defined by the class. How can I remove those?  Note: I know that I can manually add \clearpage\thispagestyle{empty} at the appropriate points of my document. However, this seems awkward and is prone to fail if I change my document later. I'm looking for automatic solutions. Inequality involving Euler-Mascheroni constant I would like to solve the following problem from 'Real Analysis and Foundations' by S. Krantz.  Let $a_n = 1+\frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} - \log{n}$ and $\gamma$ be the Euler-Mascheroni constant. Show that $$\vert a_n - \gamma \vert \le \frac{C}{n}$$ for some universal constant $C&gt;0$.  I have no clue how to solve this problem. From searching various sites it seems that $C=\frac{1}{2}$ but I am not sure with this, too. Thanks in advance. Are there any more Asgardians? Accurately simulating the lots of dice rolls without loops? OK so if your game rolls lots of dice you can just call a random number generator in a loop. But for any set of dice being rolled often enough you will get a distribution curve/histogram. So my question is there a nice simple calculation I can run that will give me a number that fits that distribution?  E.g. 2D6 - Score - % Probability  2 - 2.77%  3 - 5.55%  4 - 8.33%  5 - 11.11%  6 - 13.88%  7 - 16.66%  8 - 13.88%  9 - 11.11%  10 - 8.33%  11 - 5.55%  12 - 2.77%  So knowing the above you could roll a single d100 and work out an accurate 2D6 value. But once we start with 10D6, 50D6, 100D6, 1000D6 this could save a lot of processing time. So there must be a tutorial / method / algorithm that can do this fast? It is probably handy for stock markets, casinos, strategy games, dwarf fortress etc. What if you could simulate the outcomes of complete strategic battle that would take hours to play with a few calls to this function and some basic maths? How do you determine the size of a file in C? How can I figure out the size of a file, in bytes?  #include &lt;stdio.h&gt;  unsigned int fsize(char* file){   //what goes here? } This error message is being presented, any suggestions?     Allowed memory size of 33554432 bytes exhausted (tried to allocate   43148176 bytes) in php Every partial order can be extended to a linear ordering How do I show that every partial order can be extended to a linear ordering?      I think that I manage to prove that claim for finite set, how can I prove it for infinite set?  Thank you. Thermistor control of peltier voltage with Arduino (for DSLR chill box project) I am mildly familiar with electronics, mostly from playing around with electronics kits as a kid. That was a long time ago, though, maybe as much as 20 years ago. I am currently working on a cold or "chill" box for my Canon 5D III, so I can keep it's temperature very low, and very consistent, for doing low noise astrophotography.  I have a general design for the box, and I'm using a single 12v 5.8amp peltier (TEC) attached directly to a copper box for cooling. The box is currently insulated with extruded foam board, and the hot plate of the peltier will be cooled with a water cooler from an old computer kit.   I'm getting more ambitious with my project, however. I want thermal regulation, to maintain a consistent temperature, and I'd eventually like to get into two-stage cooling to achieve a Delta-T of closer to -55-60°C relative to ambient (the chill box will be cooling the camera, so the sensor will be warmer, probably by as much as 10°C, than the temperature of the copper plating in the box itself.) I want to have two primary modes:   Rapid cooling mode, operating the peltier at 12v or higher (max voltage is 15.4v) to quickly cool the box down to the target temperature. Regulated maintenance mode, operating the peltier at lower voltages to maintain a consistent temperature, above the maximum potential cooling the peltier can supply (for headroom as the voltage is adjusted in response to small temperature fluctuations).   I'd like to maintain temperature within 2-3°C if possible. I've looked into Arduino (and I've messed with similar things in the past), and it seems perfectly ideal for the task, with one exception: It doesn't seem to handle the kind of current I need top power a single peltier, and certainly not two.   I've done some research on how I might achieve this, but I'm falling up short on my understanding of the electronics involved. I've found a two-relay "shield" for Arduino Uno that can power two devices with up to 8amps each, and up to 30v each. That can be controlled from the Arduino itself. It seems the design of a relay uses a magnetic coil to actuate a switch that allows an independent power source to be used to power components like a motor, solenoid, or in my case a peltier. I haven't found any way of actually regulating the voltage of the relay with the Arduino, however.  So I kept investigating, and I came across some schematics that showed how to use transistors, specific mosfets to be exact, where the base was connected to an Arduino output, and the collector/emitter were connected to the power loop of whatever needed to be powered at a higher voltage (not sure about current here), and this still allowed control of the voltage.   It's been so long since I messed with any of these components, my memory is extremely rough, and I'm not quite connecting how it all works. I'd be happy with some references to complete examples of powering, and controlling the voltage of, high powered devices via an Arduino, but if anyone here can explain how all this works and why, that would be most ideal. I'd rather understand the concepts, so I can reapply them later, than just have a pattern to follow. I am trying to boot Ubuntu on my computer.  When I boot Ubuntu, it boots to a black screen.  How can I fix this? Table of Contents: Harry was never a Horcrux of Voldemort     I have heard this line on SE a lot &amp; also vaguely read about that discussion.     My question is if Harry was not Voldemort's Horcrux &amp; if all the other 6 (5 inanimate objects &amp; Nagini) were destroyed beyond repair &amp; Harry tries to Avada Kedavra Voldemort, would he die? What would happen to Voldemort's soul inside Harry.  I've asked this question keeping in mind Harry isn't a technical Horcrux but only a part of Voldemort's soul is in him. So can Voldemort resurrect from this soul?    If its a duplicate, please mention the relevant discussion link in the comments If $AB = I$ then $BA = I$ If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.    I do not understand anything more than the following.   Elementary row operations. Linear dependence. Row reduced forms and their relations with the original matrix.   If the entries of the matrix are not from a mathematical structure which supports commutativity, what can we say about this problem?  P.S.: Please avoid using the transpose and/or inverse of a matrix. Finding the missing digits of $23!$ It is given $23!=2585201xy38884976640000$. Now it is required to find the value of $x$ and $y$. I know I could find it by using divisibility rules and solving simultaneous equations. Is there any other way to solve it (without computing it by a calculator)? This question is just out of curiosity. new array.sty and colortbl.sty incompatibility? I just upgraded my texlive distribution and can no longer insert space in table rows using \\[x pt] syntax when colortbl is loaded.  It seems that recent changes to array.sty might be the problem as this is the only package to change recently.  Can anyone suggest a workaround?  MWE  \listfiles \documentclass{article} \usepackage{colortbl} \begin{document} \begin{tabular}{l}   \hline   No space below with \texttt{colortbl.sty} loaded. \\[10mm]   \hline \end{tabular} \end{document}   File versions in use:   article.cls    2014/09/29 v1.4h Standard LaTeX document class   size10.clo    2014/09/29 v1.4h Standard LaTeX file (size option) colortbl.sty    2012/02/13 v1.0a Color table columns (DPC)    array.sty    2018/04/07 v2.4g Tabular extension package (FMi)    color.sty    2016/07/10 v1.1e Standard LaTeX Color (DPC)    color.cfg    2016/01/02 v1.6 sample color configuration   pdftex.def    2018/01/08 v1.0l Graphics/color driver for pdftex The 's  feature (E:RftLW, p. 58; WGtE, p. 180) states:     [...] You ignore all class, race, spell, and level requirements on attuning to or using magic items.   Class and race requirements are relatively intuitive, additionally, since an Artificer is able to cast spells they can already attune to items that require you to be a spellcaster. But there's still some questions:   Can they use a spell scroll of a spell not on their spell list? Is this one of the four requirements ignored? What are level and spell requirements; what are some examples? I have a USB drive that I can see/access/modify/etc in Linux/Ubuntu but it doesn't show up on my Mac OSX.  I did sudo fdisk /dev/sdb (where my USB is) and then g to create a new GPT partition table, n to make a new partition, used all default settings, then w to save and exit.  Then I did sudo mkfs.fat -F32 /dev/sdb1 to format the partition with a FAT32 filesystem.  I ejected the USB and put it in my Mac and it doesn't show up / get detected.  What did I miss?