Let $$ a_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\int_{1}^{n} \frac{1}{x} d x$$ for all $n \in \mathbb{N} .$ Show that $\left(a_{n}\right)$ converges. Actually $ a_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$ is the Reimann upper sum of $f(x)=\frac{1}{x}$ on the interval $[1,n]$ relative to a partition such that  each subinterval of unit length, But I couldn't prove the convergence. Can we use the fact $\ln (n+1) \leq 1+\frac{1}{2}+\ldots+\frac{1}{n} \leq 1+\ln n$
I'm searching for a really simple and beautiful proof that the sequence $(u_n)_{n \in \mathbb{N}} = \sum\nolimits_{k=1}^n \frac{1}{k} - \log(n)$ converges. At first I want to know if my answer is OK.  My try: $\lim\limits_{n\to\infty} \sum\limits_{k=1}^n \frac{1}{k} - \log (n) = \lim\limits_{n\to\infty} \sum\limits_{k=1}^n \frac{1}{k} + \sum\limits_{k=1}^{n-1} \log(k)-\log(k+1) = \lim\limits_{n\to\infty} \frac{1}{n} + \sum\limits_{k=1}^{n-1} \log(\frac{k}{k+1})+\frac{1}{k}$     $ = \sum\limits_{k=1}^{\infty} \frac{1}{k}-\log(\frac{k+1}{k})$ Now we prove that the last sum converges by the comparison test: $\frac{1}{k}-\log(\frac{k+1}{k}) &lt; \frac{1}{k^2} \Leftrightarrow k&lt;k^2\log(\frac{k+1}{k})+1$ which surely holds for $k\geqslant 1$      As $ \sum\limits_{k=1}^{\infty} \frac{1}{k^2}$ converges $ \Rightarrow \sum\limits_{k=1}^{\infty} \frac{1}{k}-\log(\frac{k+1}{k})$ converges and we name this limit $\gamma$ q.e.d
Changing website favicon dynamically I have a web application that's branded according to the user that's currently logged in. I'd like to change the favicon of the page to be the logo of the private label, but I'm unable to find any code or any examples of how to do this. Has anybody successfully done this before?  I'm picturing having a dozen icons in a folder, and the reference to which favicon.ico file to use is just generated dynamically along with the HTML page. Thoughts? How can I boot with an older kernel version? Meaning of telegraph code in the song "Planetes" The song Planetes in Guilty Crown begins with some telegraph encoding, which I think is . Here is the . My transcription is: -•-• -••-- •-- --•-- •- --•-• •-•-- -•--•  which decodes to this:  ni-yu-ya a-i-shi-te-ru  Niyuya didn't make sense. Did I get something wrong? Why is file transfer speed decreased when copying multiple smaller files? Let's say I'm copying a portable apps directory from PC to an external HDD. An application.exe file is around 4 megabytes in size, but most other files are only tens of kilobytes in size, and most of those files take longer to copy/move than that bigger file having 4 megabytes.  Why is this? Re-use aliased field in SQL SELECT statement Concatenate several columns into one in Google Sheets If mastery of the Force is hereditary, how are the Jedi so numerous? As I was taught in school and , the T ending sound of words is unvoiced and should be pronounced with air, but recently I met a friend from the US, those aired T sounds were missing from her speaking, like "hat", "hot", "right". She speaks like /hæ/, /hɑː/,/raɪ/.   So I wonder if it's common to miss those sounds for Americans? Need to prove that $(S,\cdot)$ defined by the binary operation $a\cdot b = a+b+ab$ is an abelian group on $S = \Bbb R \setminus \{-1\}$. So basically this proof centers around proving that (S,*) is a group, as it's quite easy to see that it's abelian as both addition and multiplication are commutative. My issue is finding an identity element, other than 0. Because if 0 is the identity element, then this group won't have inverses.  The set explicitly excludes -1, which I found to be its identity element, which makes going about proving that this is a group mighty difficult. What can $E=mc^2$ do? In Windows you have a C-drive. Everything labeled beyond that is with the following letter.   So your second drive is D, your DVD is E and if you put in a USB stick it becomes F and the following drive G. And so on and so forth.  But then, what and where are A and B-drives? How to extract a number from a string Im a newbie in Apex and im developing a page using Visualforce.  My string format is always like this "xx-xx-999999". In this format I need to get only the integer. For example the string is  "RO-W1-445567". I want to get only the integer 445567 in the string. Can someone willing to share a piece of apex code in my problem.  Your help is vey much appreciated.  Thanks in advance!  Ferdie Banu Where were the Avengers during Iron Man 3? I'm having trouble restarting a 2D game when the user presses the "New Game" button.  Structure of my game:  The game has several 'screens': Opening screen, instructions screen, main game screen, and a game-over screen.  Each screen is an instance of a class: OpeningScreen class, GameScreen class, etc.  There is a class named Main. This class has three roles.   It's the JFrame (the window) of the program. It has the main method - first class to run. It has a method that change's screens.   When the program starts, this class runs. When it first runs:   Creates a new instance of itself (it's a JFrame-type class). Creates an instance of every 'screen-class'. The constructor of each 'screen-class' takes the instance of Main as an argument: (aka: GameOverScreen gameOverScreen = new GameOver(this)). Creates a new JPanel named mainPanel, and adds it to the JFrame. Any screen to be displayed will be contained in this JPanel (The screens are also JPanel-type classes).   How the changeScreen() method works:  The signature of changeScreen() is this: changeScreen(JPanel from, JPanel to).  Whenever a screen needs to be changed, the current screen that is displayed calls this method, like so: main.changeScreen(this, main.InstructionsScreen). It removes the previous screen from mainPanel and adds the new screen to mainPanel. As said, these screens are created when the program launches.  This is the method that changes screens:   public void changeScreen(JPanel from, JPanel to){      mainPanel.remove(from);     mainPanel.add(to);      mainPanel.revalidate();     mainPanel.repaint();  }   My problem:  After a game is finished, the Game class calls the changeScreen method: main.changeScreen(this, main.GameOverScreen) .  This replaces the game screen with a game over screen. Pressing the "New Game" button in the game-over screen, calls main.changeScreen(this, main.GameScreen) .  The problem: Pressing the "New Game" button in the game-over screen, indeed displays once again the instance of the Game class. But this is the old instance, so the game is not restarted. It's already finished.  I tried to do the following changes in changeScreen():  public void changeScreen(JPanel from, JPanel to){      mainPanel.remove(from);      if(to.equals(gameScreen)){         gameScreen = null;         gameScreen = new GameScreen(this);     }     mainPanel.add(gameScreen);      mainPanel.revalidate();     mainPanel.repaint();  }   This should reset the gameScreen object. But this creates a delay of about 10 seconds when pressing the "New Game" button.  What would be a good way of restarting the game?  Thank you Why couldn't Ellie's experience be corroborated by simply re-using the machine? Looking for a sci-fi movie in which a human astronaut befriends an alien on a reddish planet This probably wasn't a blockbuster back in its time as I remember the special effects didn't seem all that bombastic, but I do remember a few details.  A human astronaut lands (or crash-lands) on a prominently reddish planet. It might have been Mars, thought it might not have been. I think the atmosphere on said planet was humanly breathable (as in the astronaut does not wear his helmet).  This human astronaut needs to survive. He spots a humanoid alien, apparently also having crash-landed or otherwise in need of survival. He sees this alien consume some sort of greenish orb, apparently for nourishment. He sneaks up to the alien (or the alien craft) in an attempt to grab some of those greenish orbs. I don't remember whether he succeeds, but he does get beaten up by the alien.  The next bit of detail I remember was the human astronaut constructing some sort of shelter to protect both him and the alien from... meteor showers? The human astronaut piles up some rocks to build a hut while the alien, apparently having learned a few English words, says "shiiiiit" and "no solid". The human astronaut asks, incredulous, "No solid?" and punches the hut he just build. "See? Solid!" he proclaims, while the hut collapses a few seconds later. The alien laughs.  Later on the human astronaut is either caught by or accidentally fell into some sort of mouth-in-the-sand creature. He is saved by the alien in the process (or was it the alien who fell into the mouth-creature and was saved by the human astronaut). They witness the mouth-creature spitting out a turtle-like shell (from another local wildlife) of some sort. The human astronaut observes, and I paraphrase, "This stuff must be so tough that even [the mouth-creature] cannot digest it!" He then proceeds to build a meteor shelter from collected shells.  The last bit of detail I remember is when the alien becomes pregnant and is dying. The human astronaut, having thoroughly befriended the alien at this point, asks (and I paraphrase), "What do you mean you're pregnant? Aren't you a male?" to which the alien responds (and again I paraphrase), "For us it comes naturally, unlike for you humans whose pregnancy only comes after a brief period of pleasure". The alien then asks the human astronaut to bring the child to the alien's home planet and recite the names of the child's ancestors in front of... alien rulers, I guess. The last scene of the movie is of the human astronaut, having arrived at the alien's home planet, standing side-by-side (or perhaps hand-in-hand) with the alien child while chanting in an alien language (what I presume to be the names of the child's ancestors). Roots of polynomials in field of prime characteristic p Let $F$ be a field of characteristic $p$, and let $\alpha \in F$. Let $f \in \mathbb{Z}_p[x]$ be such that $f(\alpha) = 0$. Apparently, we have $f(\alpha^p) = 0$. This was mentioned but not proved in my lectures. Can someone explain why please?  Thanks. Proof of triangle inequality I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :) I want to communicate between several computers on my network (static Ethernet), through SSH. In order to do that I need to run ssh-add every time I log in on a specific machine, how can I do it so that it's set up once and it doesn't ask me for the passphrase every time I log in or reboot my machine?  I know that there is a way that you should add some lines to the bash_profile file, but I still need to type the password every time I reboot/log in to a specific machine.  if [ -z "$SSH_AUTH_SOCK" ] ; then     eval `ssh-agent -s`     ssh-add fi Timeout on a function call