I was studying the proof of $e$ number when I noticed something:  Why is the sequence $(1+\frac{1}{n})^{n+1}$ descending? It starts ascending with grater n but in one moment it starts descending? Why is that? On what $n$ this happens?  Edit: Let me try to rephrase the question... How come  that sequence $(1+\frac{1}{n})^n$ rises, but the sequence $(1+\frac{1}{n})^{n+1}$  falls, just because of the one more 1+1/n multiplication (power)?
Given  $$ y_n=\left(1+\frac{1}{n}\right)^{n+1}\hspace{-6mm},\qquad n \in \mathbb{N}, \quad n \geq 1. $$ Show that $\lbrace y_n \rbrace$ is a decreasing sequence. Anyone can help ?  I consider the ratio $\frac{y_{n+1}}{y_n}$ but I got stuck.
Responding to your "too localized" concerns How to use host names rather than ip addresses on home network? I've a small home network up and running and managed to set up ssh. Can anyone tell me what I need to do to refer to one computer on the network from another by host name rather than IP address (which is what I'm doing now)? I'm trying to watch for any new output of a log file. Another script (not under my control) is deleting the file then creating a new one with the same name. Using tail -f doesn't work because the file is being deleted. Is there a Google Drive client available? Is it correct to change the common structure in these phrasal verbs? I just read a book to learn English. And the topic I read is about the phrasal verbs, but a big doubt has come up to my mind.  Is it correct to change the position of the preposition (putting it with the verb instead of after the direct object)?  As in:  Common structure:   My wife backs me up over my decision to quit my job. We have to blow 50 balloons up for the party.  I need to break these shoes in before we run next week. Please help me out to get this job done.   So, would it be correct to do the following:   My wife backs up me over my decision to quit my job. We have to blow up 50 ballons for the party.  I need to break in these shoes before we run next week. Please help out me to get this job done. How do I change the GRUB boot order? I have both Windows 7 and Ubuntu installed on a shared machine. Because a lot of the non-developers use Windows, I'd like to change the boot order to make it easier for them.  Currently the boot order looks like the following:   Ubuntu 11.10 kernelgeneric *86 Ubuntu 11.10 kernelgeneric *86 (safe boot)  Memory test Memory test Windows 7 on /dev/sda6   How do I change the default order so that Windows 7 is at the top of the list?   Windows 7 on /dev/sda6 Ubuntu 11.10 kernelgeneric *86 Ubuntu 11.10 kernelgeneric *86 (safe boot)  Memory test Memory test Does magic jack work on Ubuntu? Does magic jack work with Ubuntu? I originally had my magic jack on a computer with windows XP and I decided to switch to Ubuntu but I still want Magic jack. How to delay or slowly loop commands? how to tell Latex to ignore empty lines I have shown that a smooth solution of the problem $u_t+uu_x=0$ with $u(x,0)=\cos{(\pi x)}$ must satisfy the equation $u=\cos{[\pi (x-ut)]}$. Now I want to show that $u$ ceases to exist (as a single-valued continuous function) when $t=\frac{1}{\pi}$.  When $t=\frac{1}{\pi}$, then we get that $u=\cos{(\pi x-u)}$.  With single-valued function is it meant that the function is 1-1 ?  If so, then we have that $\cos{(2 \pi-u)}=\cos{(4 \pi -u)}$, i.e. for two different values of $x$, we get the same $u$, and so for $t=\frac{1}{\pi}$, $u$ is not 1-1.  But if this is meant, how are we sure that for $t \neq \frac{1}{\pi}$ the function is single-valued? My laptop (running Win10) repeatedly "steals" the bluetooth connection for my headphones while they're connected to my phone. The only way I've been able to prevent the headphones' connection from switching to the laptop is to play some media or make a phone call immediately after reconnecting to my phone, or to unpair the headphones from the laptop.  The headphones are Plantronics Backbeat Go 2, and the specifications from the Plantronics product page give the following information for the Bluetooth entry:      Bluetooth v2.1:   A2DP 1.2, AVRCP, HFP v1.5, HSP v1.1   Thanks in advance! Is GNU's Java Compiler (GCJ) dead? How unique is the php session id How do I restore a Places folder I accidentally deleted in Nautilus? I accidentally deleted my Documents folder and didn't notice it for a couple days. Somewhere in that time, Nautilus auto-updated my list of Places so it no longer includes my Documents folder. I have since restored my Documents folder, but the Places link is still missing. How do I get it back? I want to find the probability that mean is less than 5 given that poisson distribution states it is 6 ie find p(x&lt;5|x~po(6))  Here is the actual question: Two grocers agree that the daily demand for a particular item has Poisson distribution. However, grocer A claims that the mean demand is 3 items per day, while grocer B claims that the mean demand is 6 items per day. They agree to resolve the disagreement by observing the demand on one particular day: B agrees to accept A’s claim if the observed demand is 4 or less, and A agrees to accept B’s claim if the observed demand is 5 or more. (a) Calculate the probability that A’s claim is accepted when, in fact, B’s claim is correct Why does my ArrayList contain N copies of the last item added to the list? I tried logging in with a Google account, but I got the following error: What are the practical implications of a name change for international travel and activities abroad? I am considering a name change, which seems to be a relatively straightforward action, when one remains in the same country: certain documents are replaced with new ones bearing the new name, and others are presented together with the name change document.  However, I travel internationally multiple times per year, and I often change temporary residence from one country to another. So I often deal with immigration and similar things. Furthermore, I have a lot of things going on like work and studies.  Quite obviously, if I change my name, I would often have to produce a proper translation of the name change document, which may include notarized translation thereof with apostille or, for example, traduction assermentée in France.  However, it would be great to hear about experiences of people, who have dealt with this issue first-hand, so I can weight the disadvantages against advantages of this correctly. In particular, I would like to know if anyone has encountered unforeseen difficulties with this, and how people rate the overall difficulty of such experiences. Booting Ubuntu Failure : error: attempt to read or write outside of disk 'hd0' I have installed Ubuntu 12.10 in a Western Digital external hard drive (320GB). This is a complete installation, not a live USB.  When I plug it in my HP desktop I go to the BIOS settings and boot off the hard drive, everything work perfectly as it should. Now this works on every single computer and laptop in my house (all HP), except for ONE. My HP ProBook 4530s.  When I select to boot of the USB I get the message:  error: attempt to read or write outside of disk 'hd0'   Now, I have removed the hard drive from my laptop and the external drive is the ONLY drive plugged in. Below is a screenshot of the message on the screen. After the message I navigate to ls / (as shown below):    After here I try to acces other folders under ls /, for example, I try to go to ls /boot to get to the grub folder. Then I get the same message as before: as shown below:    grub rescue&gt; ls /boot  error: attempt to read or write outside of disk 'hd0' grub rescue&gt; _   The only folders I can access without getting the message again are /home, /run and /usr.   So how do I:   Boot Ubuntu from GRUB2 (this screen) manually Set to automatically boot Ubuntu If possible an explanation for this problem   Thanks! What's the protocol for requesting a tag warning? On a particular site, RPG Stack Exchange, we've decided on our meta on a couple of tags that should have tag warnings, and on what those tag warnings should say. I am  on the site in question.  What should we do to get the tag warnings implemented? Do we do any/all of the following?:   start a [feature-request] on our meta for the tag warnings? raise a CM escalation? contact staff via the contact form? ping a staff member in the Teacher's Lounge? anything else?