I just can't get my head wrapped around a certain step in a proof that the sequence that approaches $e$ is monotonically increasing. The proof starts as follows:  We want to show that: $x_{n+1} \geq x_n \iff \\ \big(1 + \frac{1}{n+1}\big)^{n+1} &gt; \big(1 + \frac{1}{n}\big)^{n} \iff  \\ \big(\frac{(n+2)n}{(n+1)^2}\big)^n &gt; \frac{n+1}{n+2} \iff ... \\ $  Now I have trouble understanding how we get from line 2 to line 3, anyone have a simple hint that can help me?  This question is not a duplicate because it asks for one certain step in an approach that is not covered in the linked question.
Show that $U_n:=\left(1+\dfrac{1}{n}\right)^n$, $n\in\Bbb N$, defines a monotonically increasing sequence.   I must show that $U_{n+1}-U_n\geq0$, i.e. $$\left(1+\dfrac{1}{n+1}\right)^{n+1}-\left(1+\dfrac{1}{n}\right)^n\geq0.$$  I am trying to go ahead of this step.
Hardware Enablement Stack (HWE) out of support Can there be a notification for bounties started on questions you've asked or answered? How can I cast int to enum? Is the W statistic output by wilcox.test() in R the same as the U statistic? I've recently been reading about the Mann-Whitney U test. It turns out that to carry out this test in R you actually need to run a Wilcoxon test!  My question: is the W statistic of wilcox.test in R identical to the U statistic? Do you capitalize both parts of a hyphenated word in a title? Do you capitalize both parts of a hyphenated word in a title?  "My Ex-Wife Hates Me"  or  "My Ex-wife Hates Me" PythonTEX not working on current Overleaf compiler Is there a way to make PythonTEX compile as of today in Overleaf V2 ?  \documentclass[]{article} \usepackage[utf8]{inputenc} \usepackage{pythontex}  \begin{document} \begin{pyverbatim} def calc_ellipticity(theta, delta):     theta = np.deg2rad(theta)     psi = np.arcsin(np.sin(2*theta) * np.sin(delta))/2     return np.tan(psi) \end{pyverbatim} \end{document}   Compiling gives : ? PythonTEX ? Which singular names ending in “s” form possessives with only a bare apostrophe? I have a sword that currently rotates around a pivot point thats off of the actual object. Im new to blender and im not sure how i would go about changing it so that the sword rotates around the handle. Any help would be appreciated! The  (or conjecture), suggests that black holes can be entirely described by their mass, angular momentum and charge. All other details of the BH formation are lost.  Is there a simple way of understanding why you would be able to tell the difference between a BH made from protons and one made from electrons, but not between one made from matter rather than anti-matter?  i.e. What is special about charge, and why aren't lepton number, baryon number, strangeness etc, also black hole properties? remove divs and spans from view output In my page.tpl.php I'm doing this to display one of my view:  print views_embed_view('teaserview', 'default');   That gives me the following html output:  &lt;span thmr="thmr_10"&gt; &lt;div class="view view-teaserview view-id-teaserview view-display-id-default view-dom-id-e051ba535ce9aac6cd9a0d564b3dbb29 contextual-links-region"&gt;     &lt;div class="contextual-links-wrapper"&gt;         &lt;span thmr="thmr_16"&gt;         &lt;ul class="contextual-links"&gt;             &lt;li class="views-ui-edit first last"&gt;&lt;span thmr="thmr_17"&gt;&lt;a href="/kbi/admin/structure/views/view/teaserview/edit/default?destination=node/3"&gt;Edit view&lt;/a&gt;&lt;/span&gt;&lt;/li&gt;         &lt;/ul&gt;         &lt;/span&gt;     &lt;/div&gt;     &lt;div class="view-content"&gt;         &lt;span thmr="thmr_15"&gt;         &lt;div class="views-row views-row-1 views-row-odd views-row-first views-row-last"&gt;             &lt;span thmr="thmr_14"&gt;             &lt;span thmr="thmr_11"&gt;&lt;span thmr="thmr_12"&gt;&lt;span thmr="thmr_13"&gt;&lt;img src="http://127.0.0.1/kbi/sites/default/files/teaser_2.jpg" width="1029" height="308" alt="" /&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;         &lt;/div&gt;         &lt;/span&gt;     &lt;/div&gt; &lt;/div&gt; &lt;/span&gt;   But the only thing that I really need is  &lt;img src="http://127.0.0.1/kbi/sites/default/files/teaser_2.jpg" width="1029" height="308" alt="" /&gt;   Is there a possibility to modify the views_embed_view call so that I get only the needed data?   There would be the possibility to modify all the views .tpl.php files but I hope there's an easier way to do this.   I also read about the semantic views module but I didn't understand how to configure my view with it so that i get only the needed markup as output. it didn't seem to have any effect. VirtualBox 5.0.40 crashes the whole host since update to kernel 4.13.0-26-generic When I signed up I received a welcome message specifically indicating I should take the tour to earn my first badge:     Clicking the message takes me directly to the .  However, as   , the 'Informed' badge is not active here on meta.  Shouldn't the wording of the welcome message be altered if a user signs up on a site which does not use this badge? What do "watched tags" and "ignored tags" do? Why define an anonymous function and pass it jQuery as the argument? The arrival of Minecraft Beta 1.5 brought weather conditions. However, it can at times be inconvenient when you're trying to screenshot something and show it off (as it reduces lighting and obstructs your view slightly). Is there anyway to turn weather off?  Alternatively, is there a way to speed up the duration of weather or "skip" it? I have Solaris 8 OS installed in French.  Is it possible to switch from French to English without reinstalling the OS ? If yes how ?  When locale -a i have this :  POSIX common en_US.UTF-8 C iso_8859_1 en_CA en_CA.ISO8859-1 en_US en_US.ISO8859-1 en_US.ISO8859-15 en_US.ISO8859-15@euro fr_CA fr_CA.ISO8859-1 iso_8859_15 th th_TH th_TH.TIS620   Even when I try to edit /etc/default/init with :  LANG=en_US.ISO8859-15 LC_COLLATE=en_US.ISO8859-15 LC_CTYPE=en_US.ISO8859-15 LC_MESSAGES=en_US.ISO8859-15 LC_MONETARY=en_US.ISO8859-15 LC_NUMERIC=en_US.ISO8859-15 LC_TIME=en_US.ISO8859-15 LC_ALL=en_US.ISO8859-15   and reboot : nothing happens.  To verify the result i always use date and when i reboot : i also have the date in french.  P.S : I have to do this configuration in a computer which isn't here in my office, so i have to this manipulation in a an another computer here in my office and if it works, i'll do it to the other computer.  So the computer here is configured in English and i have to switch it to French. Here is the content of my ~./dt :  C                     errorlog              sessionlogs Desktop               errorlog.old          sessions Trash                 errorlog.older        simulateur.dtwmrc appconfig             help                  startlog appmanager            icons                 startlog.old asr.dtwmrc            outils.dtwmrc         startlog.older bin                   palettes              tmp bookmarks             sdtfonts              types dtwmrc                sdtpcaction.conf      wsmenu en_US.ISO8859-15      sdtpcaction.conf.bak   Thanks What does this notification icon mean? Samsung s7 Spyder - touchpad scroll does not work I use the scroll function with my mouse and it works fine. But either in script, or in terminal sections of Spyder, I can't use the touchpad for scrolling. I do not experience any problems with my touchpad in other programs like Word, Chrome etc. Is there any option for this in Spyder, specific for touchpad?  I am using Lenovo Legion Y520, Windows 10. Prove that:   $$p \,\,\left|\, {p \choose k} \right., \quad 0&lt; k \lt p$$   if $p$ is prime.   how to prove that with direct proof? For example, when I type ping 8.8.8.8 &amp;, why can we see ping process working? And when I type find / -name '*test*' &amp; , it's also seen on monitor.  Why is that? Is it not really in background?