Is $ 2^n + n &lt; 2^{(n+1)}$ when $ n &gt; 2 $  How can I prove this? I need it in order to prove a language is context free
Prove that  $ n &lt; 2^{n}  $ for all natural numbers $n$.  I tried this with induction: Inequality clearly holds when $n=1$. Supposing that when $n=k$, $k&lt;2^{k}$. Considering $k+1 &lt;2^{k}+1$, but where do I go from here?  Any other methods maybe?
How can I gear my follower post 2.0? I'm a little confused as to how I'm supposed to gear up my follower since the installation of smart loot in Loot 2.0. My follower still has most pre loot 2.0 items because I have yet to find comparables that are rolling with the right affixes.   To be clear, I play a DH and so I used the Templar as my tank. I very rarely find items that roll with Strength because smart loot sees my DH and rolls Dex for me instead. With the use of the Enchantress I thought it would get better because I could replace Dex with Strength...but that's not true. It seems as though the Enchantress will only roll Dex for me too.   With the auction house down for good, no outlet for trading, and the installation of smart loot, it seems that gearing a follower has become almost impossible unless I want to jump on another character that uses strength and get items that way. I want to find items that I can use while I'm playing. How do I gear my follower post Loot 2.0? Do I have any options other than character jumping? degree of freedom of a rigid body 5 or 6? I'm confused here. I have a three particle (rigid) system. What would be the degree of freedom? I found out five. 3 coordinates for center of mass and 2 for describing orientation. But we have only three constraints, i.e. three equations that reduce 9 coordinates by 3, 9 - 3 = 6, which gives 6 degrees of freedom?? Did I miss something above? Can I combine extension tubes and macro filters at once to give better macros? Is there a easy way to test ubuntu on new notebooks before buying it? FFmpeg get duration of video file without meta data I got a video file which has no meta data.  If I do this for example:  ffmpeg -i test.m2v   I get these values:  Duration: N/A, bitrate: N/A   Is there still a way to get the duration of the video / .m2v file ?  Edit:  The full console output:  ffmpeg version 2.8.4 Copyright (c) 2000-2015 the FFmpeg developers   built with gcc 5.2.0 (GCC)   configuration: --enable-gpl --enable-version3 --disable-w32threads --enable-avisynth --enable-bzlib --enable-fontconfig --enable-frei0r --enable-gnutls --enable-iconv --enable-libass --enable-libbluray --enable-libbs2b --enable-libcaca --enable-libdcadec --enable-libfr eetype --enable-libgme --enable-libgsm --enable-libilbc --enable-libmodplug --enable-libmp3lame --enable-libopencore-amrnb --enable-libopencore-amrwb --enable-libopenjpeg --enable-libopus --enable-librtmp --enable-libschroedinger --enable-libsoxr --enable-libspeex --enab le-libtheora --enable-libtwolame --enable-libvidstab --enable-libvo-aacenc --enable-libvo-amrwbenc --enable-libvorbis --enable-libvpx --enable-libwavpack --enable-libwebp --enable-libx264 --enable-libx265 --enable-libxavs --enable-libxvid --enable-lzma --enable-decklink --enable-zlib   libavutil      54. 31.100 / 54. 31.100   libavcodec     56. 60.100 / 56. 60.100   libavformat    56. 40.101 / 56. 40.101   libavdevice    56.  4.100 / 56.  4.100   libavfilter     5. 40.101 /  5. 40.101   libswscale      3.  1.101 /  3.  1.101   libswresample   1.  2.101 /  1.  2.101   libpostproc    53.  3.100 / 53.  3.100 Input #0, mpegvideo, from '.\Test.m2v':   Duration: N/A, bitrate: N/A     Stream #0:0: Video: mpeg2video (Main), yuv420p(tv), 720x576 [SAR 64:45 DAR 16:9], max. 7000 kb/s, 25 fps, 25 tbr, 1200k tbn, 50 tbc Where to place my local website starting with the 2.4.7 version of apache2? I recently installed Ubuntu 14.04, then I installed lamp-server and placed my webpages in /var/www directory, but when I opened localhost in the browser there was nothing. I think that happened because Zend updated Apache.  Whatever the reason may be, I want to know where to keep my php files so that I can access them from my browser. I have to prove this : "$n^7-n$ is a multiple of 7". This is what I have done this so far :  $P(n):n^7-n$ On putting $n=1,$ $P(1):1^7-1=0$, which is a multiple of 7. So, $P(1)$ is true. Let $P(k)$ be a multiple of 7. So, $k^7-k$ is a multiple of 7. So, $k^7-k=7m$, where $m$ is any natural number. Let $P(k+1)$ be true, Hence, $(k+1)^7-(k+1)$ is a multiple of 7.  Now, how do I evaluate $(k+1)^7$ without using the binomial theorem ? Or is there some other way to do it ? STM32 send from UART3 and receive on UART2 I thought to create a simple test to send and receive a buffer via UART. I succeeded in sending bytes from my PC to the nucleo bord and receiving them in blocking and non-blocking mode. But for some reason I can't succeed to send data from UART3 and receive the bytes on UART2.  I have a nucleo-f103rb and have the following pins setup:   UART2: tx=PA2;rx=PA3 UART3: tx=PB10;rx=PB11   And then the following code:    while (1)   {     /* USER CODE END WHILE */     /* USER CODE BEGIN 3 */     if (HAL_UART_Transmit(&amp;huart3, send, 2, 1000) == HAL_OK)     {         if (HAL_UART_Receive(&amp;huart2, receive, 2, 1000) == HAL_OK)         {             HAL_GPIO_TogglePin(GPIOA, LD2_Pin);         }     }   }   All the boiler plate is generated using cube mx. So the UART 2 and 3 are all default, same for the button pins.    I connected the transmit pin of uart3 to the receiving pin of uart2 I connected the receive pin of uart3 to the transmitting pin of uart2 (while not used in this trivial example though)   What am I missing here?  Update  As comments and answers suggested, I shouldn't be using the blocking mode approach. So I rewrote it slightly to rely more on interrupts.  I attached a callback to the nucleo blue button  void HAL_GPIO_EXTI_Callback(uint16_t GPIO_Pin) {     if (GPIO_Pin == B1_Pin)     {         HAL_UART_Transmit_IT(&amp;huart3, (uint8_t *)send, 2);     } }   And I added a callback for receiving stuff over uart  void HAL_UART_RxCpltCallback(UART_HandleTypeDef *huart) {   /* Prevent unused argument(s) compilation warning */   UNUSED(huart);    /* NOTE : This function should not be modified, when the callback is needed,             the HAL_UART_RxCpltCallback can be implemented in the user file    */       // Toggle the pin to see some proof that I received something      HAL_GPIO_TogglePin(LD2_GPIO_Port, LD2_Pin);       // Resubscribe uart2      HAL_UART_Receive_IT(&amp;huart2, (uint8_t *)receive, 2); }   In the main function I call the interrupt so that the callback is active.  while (HAL_UART_Receive_IT(&amp;huart2, receive, 2) != HAL_OK);  __NOP(); while (1) { }   Again, when debugging through I do end up in the button released callback, but the callback on the uart receive isn't triggered. When I send stuff from the PC, it does. So it seems that sending stuff via uart3 isn't working. So I guess that must be the wiring.     But I really can't see what I am missing here, the pins are clearly defined in the pinout here: .  I hope the photo is clear enough, but this is how I connected them. What time does sound good to you?     What time sounds good to you?    Is there a difference? What can I do to lift a 10 year ban for making a mistake in my UK visa application? I am a Nigerian citizen.  In 2007 I was twice denied a visa to the UK.  Recently I was filling out an application for another UK visitor visa and by mistake I answered no to the question whether I had ever been denied a UK visa, instead of yes.  When reviewing my application, the visa officer found out about the previous visa rejections and as a result I got a 10 year ban on travel to the UK.  What can I do to lift my 10 year ban?   Will this ban affect my chances of obtaining a visitor visa to the USA? Smoothness constraint of wave function Is there anything in the physics that enforces the wave function to be $C^2$? Are weak solutions to the Schroedinger equation physical? I am reading the beginning chapters of Griffiths and he doesn't mention anything. Do SSD's really last 285 years? Some Sold State Disks (SSD) have a  (MTBF) of 2,500,000 hours. If you divide that by the rough number of hours in a year you get 2500000÷(365×24) ~= 285 years before there is a problem.  I know that the first generation SSD's used to have a short life - but now the estimated time before they fail seems absurdly high. Most warranties only cover them for 1-3 years it seems.  How long are the modern, SATA III drives supposed to last? How to hide terminal output when executing a command? "approach to address" or "approach to addressing"? Depending on our point of view, we offer different approaches to address(ing?) a same issue.   Should I use &mdash;ing in this context, and if yes, why? Bash script copy file to user's (wildcard) home dir Can all fundamental forces be fictitious forces? iOS crash log catch, debug info.. Catch and send via email to the Dev team Why Process Builder doesn't work on Delete Scenario? YA book about a young powerful female caster attending a school that trains in magical arts and duels I read a young adult novel around the turn of the century. This story features a young female protagonist that is very naturally gifted in magic. She is attending a  to learn mastery and control over her gifts.  A lot of her school revolves around magic dueling. The participants are taught to extend their inner aura like a second magical skin, possibly referred to as creating a shield, or . The user can increase the strength of the shield by focusing harder on it, but doing so means you have less ability to be offensive. The amount of energy needed to create a shield was half the energy a foe would need to focus at you to break it.  Ex: Mage A and B are equally matched. Mage A focuses half his power into a shield. Mage B would then need to focus 100% of his power to break A’s shield.  Another aspect I remember is that when dueling, a school master would extend a personal shield on to one of the combatants as a sort of secondary shield that goes underneath the combatants. When one mage broke the others mages personal shield, the school masters shield protects the losers life from harm, and the breaker of the shield is declared the winner.  I distinctly remember one match in which the protagonist is having a duel with another strong student. She creates a wind storm around herself that picks up the sand from the arena floor and makes her impossible to see and thus makes it hard for her opponent to focus on her shield to break it. She then lashes out at her opponent with incredibly powerful strikes with the final strike being so hard it shatters the opponents shield and then breaks the school masters shield as well and stops just shy of killing her dueling partner.  Any assistance on this one? I think my daughter would love to read it. What can I do to increase passive HF RFID read range? I am working on a prototype, one function of which is to detect whether an object is "in range" by a few cm. Right now I'm just using an Arduino and a cheap  that operates at 13.56 MHz:    I've successfully gotten it to detect the tag, but only at a distance of about 3-4 cm when the tag is oriented parallel to the antenna.  According to my  , the HF band allows read ranges of between 10 cm and 1 m for passive tags, depending on factors like antenna size, tag size and orientation, and transceiver power.  I realize this inexpensive reader is not up to task as-is, but I'm interested in building something that achieves read ranges of 15-20 cm.  Some criteria:   No need to read multiple tags; only whether one tag is in range or not. No need to write to the tag. Cannot use an active tag.   I'm not experienced with RFID reader construction (yet) but am interested in knowing what I should pursue.  Here are some things I've considered:   Ditch the embedded antenna on the cheap RFID reader and construct my own, larger antenna. I'd need to see if the onboard  is up to the task of providing additional power. Look for a different IC around which to build a reader from scratch. Find a ready-made solution instead of trying to build one (more cost-effective?). Use something other than RFID for simple proximity detection.   I am experienced with microcontrollers and basic electronics, and have made audio and lighting projects, but this is a first with RFID. Since I don't need to read multiple tags, which RFID is suitable for, I've considered maybe some sort of magnet and hall-effect sensor, but the range does not seem adequate. I considered using an ultrasonic proximity detector, but the application calls for detecting the tagged object while ignoring untagged ones (ultrasonic would produce false positives on untagged objects). I've considered some sort of reflective surface placed on the object, and detecting reflected light, but the object orientation is not reliable enough to maintain alignment.  So, in a nutshell, What modifications can I make to my existing reader, or what alternate technology should I pursue to achieve 15-20 cm detection of a passively tagged object?  I plan on connecting whatever proximity detector is used to a microcontroller using SPI.