Constructing a contradiction to prove limit of a recurrence relation
Verify proof that sequence converges
I want to prove with mathematical induction that:  $$\sum_{i=1}^n i \cdot i! = (n+1)! - 1$$  So in the first step we define $n = 1$:  $$\sum_{i=1}^1 i \cdot i! = 1 \cdot 1! = 1 = 2! - 1$$  In the second step we define $n = n + 1$:  $$\sum_{i=1}^{n+1} i \cdot i! = \sum_{i=1}^n i \cdot i! + (n+1)(n+1)!$$ $$\implies (n+1)! - 1 + (n+1)(n+1)!$$  This can now be simplified to $(n+2)(n+1)! - 1$ but I do not see how...  We could do $(n+1)!(-1 + (n+1)) = (n+1)! \cdot n$ but still this will not be the same as above.  What did I miss?  Bodo Is it common to use "where" after "a time" as a relative? I heard the following comment of Mr. Marco Rubio about Mr. Donald Trump on CBS’s “Face the Nation” in a clip in February 29th AP Radio News:  “There is never gonna be a time where the Republican Party rallies around and says you have to get out or anyone has to get out for purposes of rallying around Donald Trump. Donald Trump is not Republican Conservative. Donald Trump is trying to pull off the biggest scam in American political history.”  I thought I misheard “There is never gonna be a time where –“ for “There is never gonna be a time when –“ at first, because I was in understanding that time should be trailed by “when” as a relative noun or adverb. An English Japanese dictionary at hand (Readers English Japanese Dictionary published by Kenkyusha) provides “where” singly as a relative to refer the ‘place.’ However, my lookup of the following two sources endorsed that I heard it right:  Here is what is never going to happen in this race. There's never going to be a time where the Republican Party rallies around and says you have to get out or anyone has to get out for purposes of rallying around Donald..  “Here is what is never going to happen,” Rubio continued. “There's never going to be a time where the Republican Party rallies around and says you have to get out or anyone has to ...   Is it common, at least not uncommon to use “where” after “time” as a relative? How can I control a servo motor with the PWM output of a microcontroller? I wanted to control the direction of servo motor with a switch. I am using an Atmega32.  Outputs are PIND4 and PIND5 and the input is PINA3, but the two servo motors turn only in 1 direction.  #ifndef F_CPU  #define F_CPU 1000000UL   #endif  #include &lt;avr/io.h&gt; #include &lt;util/delay.h&gt; #include &lt;avr/interrupt.h&gt;   int main(void) { DDRD= 0xFF ; DDRA=0x00; PORTA =0x00; // FAST PWM TCCR1A |= 1&lt;&lt;COM1A0 | 1&lt;&lt;COM1A1 | 1&lt;&lt;WGM11 |1&lt;&lt;COM1B0|1&lt;&lt;COM1B1 ; TCCR1B= 1&lt;&lt;WGM12|1&lt;&lt;WGM13 ; ICR1 = 19999 ;  TCCR1B |=1&lt;&lt;CS10 ; while (1) {    if(PORTA==0x08) //if pina3 is 1    {     OCR1A = ICR1 -2000 ;   //turn servo 1     _delay_ms(1000);          OCR1B=ICR1-2000;    //turn servo 2     _delay_ms(1000);     }     if(PORTA==0x00)// if pina3 is 0     {       OCR1A = ICR1 -900 ; //turn servo 1 in the other direction       _delay_ms(1000);               OCR1B=ICR1-900;     //turn servo 2 in the other direction       _delay_ms(1000);      } }       return 0 ;  } How do you configure tomcat to bind to a single ip address (localhost) instead of all addresses? How to increase JAVA Heap Size? Low output volume Choice of $q$ in Baby Rudin's Example 1.1 First, my apologies if this has already been asked/answered.  I wasn't able to find this question via search.  My question comes from Rudin's "Principles of Mathematical Analysis," or "Baby Rudin," Ch 1, Example 1.1 on p. 2.  In the second version of the proof, showing that sets A and B do not have greatest or lowest elements respectively, he presents a seemingly arbitrary assignment of a number $q$ that satisfies equations (3) and (4), plus other conditions needed to show that $q$ is the right number for the proof.  As an exercise, I tried to derive his choice of $q$ so that I may learn more about the problem.  If we write equations (3) as $q = p - (p^2 - 2)x$, we can write (4) as   $$ q^2 - 2 = (p^2 - 2)[1 - 2px + (p^2 - 2)x^2]. $$  Here, we need a rational $x &gt; 0$, chosen such that the expression in $[...]$ is positive.  Using the quadratic formula and the sign of $(p^2 - 2)$, it can be shown that we need   $$ x \in \left(0, \frac{1}{p + \sqrt{2}}\right) \mbox{ for } p \in A, $$  or, for $p \in B$, $x &lt; 1/\left(p + \sqrt{2}\right)$ or $x &gt; 1/\left(p - \sqrt{2}\right)$.  Notice that there are MANY solutions to these equations! The easiest to see, perhaps, is letting $x = 1/(p + n)$ for $n \geq 2$.  Notice that Rudin chooses $n = 2$ for his answer, but it checks out easily for other $n$.  The Question:  Why does Rudin choose $x = 1/(p + 2)$ specifically?  Is it just to make the expressions work out clearly algebraically?  Why doesn't he comment on his particular choice or the nature of the set of solutions that will work for the proof?  Is there a simpler derivation for the number $q$ that I am missing? Can a Paladin take levels of other classes? In 3.5 D&amp;D,  have severe restrictions on multiclassing:     Like a member of any other class, a paladin may be a multiclass character, but multiclass paladins face a special restriction. A paladin who gains a level in any class other than paladin may never again raise her paladin level, though she retains all her paladin abilities.   Because Pathfinder is based on 3.5 D&amp;D, do Pathfinder paladins face the same restriction? That is, if a Paladin multiclasses to say, a fighter, can she take another level of Paladin later on? Is dimensionality reduction almost always useful for classification? What happens if 2 separate abilities require you to sacrifice a creature? I have a machine running a couple of vagrant VM. The problem I have is that sometimes I forget to shutdown those VM before I shutdown or reboot my machine. Because of that my machine get stuck with this message: waiting for vboxnet0 to become free  I searched about solutions and I found this page :    I tried what they for shutdown, but it doesn't work.  I wrote an sh file for that command:  #!/bin/bash  cd ~/workspace/git/mediaservice vagrant halt   any suggestions? Fantasy/scifi combo - band of travelers who must get from "sphere" or domain to domain - each geographic area is ruled by a magician - it turns out later these are people who are unwittingly tapping into the computer grid they all live on to alter their own reality? Moon photography Hi I am a photography hoobyist. At the moment I am using a Nikon D3100 with the standard kit lens (18-55mm). I am interested in taking photos of the moon. I am debating whether to buy a 55-200mm lens or a 55-300mm lens. Which one would be better?  P.S I am on a tight budget and I am looking for a lens for personal use, not commercial.  Would appreciate any answers!  Thank you! Amazon RDS I/O requests - 1 query = 1 billable I/O? I have an InnoDB database that I'd like to migrate onto Amazon RDS.  My current implementation, hosted on my own server, shows roughly 8M queries per month.   say that  I/O Rate    $0.10 per 1 million requests  Does 1 I/O = 1 query?  ie, will I be billed $80/mo for this amount of usage + the RDS fees? So I just started using linux/ubuntu today and I've been getting problems with my wifi, ive tried the solutions on this page:  When im really close to my router, its fine, i can see wifi signals, but when i go just a few steps back it disappears right away. Im using the latest version of ubuntu 17.10  Any suggestions would be nice! :) Prove that ${\sqrt {n} }^{\sqrt {n+1}} > {\sqrt {n+1}}^{\sqrt {n}}$. Page number position How does DC work with natural 20? How long does it take to unlock another frame by playing one of the base frames? I'd like to get an advanced frame in Firefall. I've been told that I can use "Pilot Tokens" to unlock other frames without having to pay money.  How soon would I be able to unlock an advanced frame? Do I have to level a base frame all the way to level 40? More than once? Let us say you are in an inertial reference frame with a circular planar disk. If you take your meter measuring rods (or perhaps tape measure) you can find the diameter and circumference of the disk. If you divide the circumference by the diameter, you will get exactly $\pi$. Now you start rotating the disk. A book I have claims that now the ratio will be different (lets call it $\pi_\circ$ to avoid ambiguity.) This has caused to reevaluate my understanding of special relativity (this is a set up to general relativity, but the problem itself only requires the special theory.) My problem is that, sure, the measuring rods (or tape) would contract on the circumference, but wouldn't the circumference contract a similar amount, therefore giving a measurement of $\pi$ in every reference frame according to your own measuring rods. Of course, this reasoning goes on to make me question how I understood all of special relativity up this point.