How to show $\frac{2x}{2+x}&lt;\log(x+1)$ for $x&gt;0$   Without differentiating, more elementary (it looks then more complicated but OK)  $\log(x+1)=\int\limits_1^{x+1}\frac 1udu$ and $\frac{2x}{2+x}=x\frac{1}{ 1+\frac x2}$ hence   $\log(x+1)-\frac{2x}{2+x}=\int\limits_1^{x+1}\frac 1u-\frac{1}{ 1+\frac x2}du=\int\limits_1^{x+1}\frac{1+\frac x2-u}{u\left(1+\frac x2\right)}du=\int\limits_1^{\frac x2+1}\frac{1+\frac x2-u}{u\left(1+\frac x2\right)}du-\int\limits_{\frac x2+1}^{x+1}\frac{u-\left(1+\frac x2\right)}{u\left(1+\frac x2\right)}du$  The numerator of the integrand in the first integral is positive and negative in the $2$nd. But for the first one the denominatior is smaller, so the fraction is always bigger than its corresponding fraction in the second one, and so the difference is always positive  is this correct, do you have an alternative proof
EDIT:  My question was poorly worded.   I wasn't asking about showing $\ln(1+x) &gt; \frac{2x}{2+x}$ for $x&gt;0$.   What I wanted to know is why the lower bound provided by $ \frac{2x}{2+x}$ was so tight for small positive values of $x$. This is addressed in robjohn's answer.
How do I change the location of macOS screen captures? As a developer, I take a lot of screen captures. I often take screen shots to share with Stack Exchange members in chat.  The problem is my Desktop is so cluttered with screen captures that they're now overlapping! I'd like to save all screenshots in a dedicated folder or directory.  In macOS, how do I change the default location where screen captures are saved?  These are the screen shots taken with CommandShift3 or CommandShift4 keyboard shortcuts.  I would prefer a solution which doesn't require installing any software. Can I watch the progress of a `sync` operation? What can cause a question to be bumped? What can cause a question to be "bumped" to the first page in the Active tab of a Stack Exchange site? (Changing the "Last Active" date.)  . How can I increase my Vault Dweller's Happiness? How to determine which program is making a certain sound in windows 7? I'm experiencing an annoying "female robotic" sound effect saying "loading complete" at uncertain interval in windows 7.  I want to shut down that noise for good. To do that, I have to determine which program is making this sound. I tried to open the "sound mixer" window, only to find to program last made any sound, one is official GTalk client, one is system sound. I tried to close GTalk and the sound still existed.  I started experiencing this issue not long after I installed a "driver" program for Rapoo E9198 wireless keyboard. I am working on Ubuntu 12.04 64bit.  I want to add "Open terminal here" to Nautilus context or right-click menu but it tries to download 32bit version from Internet. After burning Ubuntu 12.04 to a CD, I tried booting and got the message:  This kernel requires the following features not present on the CPU: pae  Unable to boot - please use a kernel appropriate for your CPU.   What does this mean? I have installed Ubuntu alongside Windows 8.1. But the wifi is not working in Ubuntu (perhaps the drivers are not present), which worked fine in windows. Also I cant connect with Ethernet since my laptop does not have an Ethernet port. So is there any way for me to connect to the internet in ubuntu? Edit: Output of the terminal command  lspci -knn | grep Net -A2:  vikram@vikram-Inspiron-11-3148:~$ lspci -knn | grep Net -A2 01:00.0 Network controller [0280]: Broadcom Corporation BCM43142 802.11b/g/n [14e4:4365] (rev 01)     Subsystem: Dell Device [1028:0018]     Kernel driver in use: bcma-pci-bridge vikram@vikram-Inspiron-11-3148:~$    I did what Pilot6 suggested. Here is the output:    vikram@vikram-Inspiron-11-3148:~$ sudo dpkg -i *.deb [sudo] password for vikram:  Selecting previously unselected package bcmwl-kernel-source. (Reading database ... 163804 files and directories currently installed.) Preparing to unpack bcmwl-kernel-source_6.30.223.141+bdcom-0ubuntu2_amd64.deb ... Unpacking bcmwl-kernel-source (6.30.223.141+bdcom-0ubuntu2) ... Selecting previously unselected package dkms. Preparing to unpack dkms_2.2.0.3-1.1ubuntu5.14.04_all.deb ... Unpacking dkms (2.2.0.3-1.1ubuntu5.14.04) ... Setting up dkms (2.2.0.3-1.1ubuntu5.14.04) ... Setting up bcmwl-kernel-source (6.30.223.141+bdcom-0ubuntu2) ... Loading new bcmwl-6.30.223.141+bdcom DKMS files... First Installation: checking all kernels... Building only for 3.16.0-30-generic Building for architecture x86_64 Building initial module for 3.16.0-30-generic ERROR (dkms apport): kernel package linux-headers-3.16.0-30-generic is not supported Error! Bad return status for module build on kernel: 3.16.0-30-generic (x86_64) Consult /var/lib/dkms/bcmwl/6.30.223.141+bdcom/build/make.log for more information. modprobe: FATAL: Module wl not found. update-initramfs: deferring update (trigger activated) Processing triggers for man-db (2.6.7.1-1ubuntu1) ... Processing triggers for initramfs-tools (0.103ubuntu4.2) ... update-initramfs: Generating /boot/initrd.img-3.16.0-30-generic vikram@vikram-Inspiron-11-3148:~$ Are the groups $\mathbb{C}$ and $\mathbb{R}$ isomorphic? Generating Random Passwords When a user on our site loses his password and heads off to the Lost Password page we need to give him a new temporary password. I don't really mind how random this is, or if it matches all the "needed" strong password rules, all I want to do is give them a password that they can change later.  The application is a Web application written in C#. so I was thinking of being mean and going for the easy route of using part of a Guid. i.e.  Guid.NewGuid().ToString("d").Substring(1,8)   Suggesstions? thoughts? I have started a long process through a terminal. Is it possible to make the Ubuntu terminal make a sound once the process is complete? This way, I don’t need to keep checking, but will instead be notified through a sound. A formula for Perspective measurement If I have a photograph of a rectangular object whose image recedes with perspective, is it possible to work out the scale of measurements along that object?  For example: I have a photograph of a (metric) ruler.  The ruler is graduated in centimetres as shown, 30 cms.  Is it possible to work out the distance between each centimetre?  The screenshot shows the photograph.  The representation on the Picture Plane shows the nearest edge of the ruler to be 50mm and the farthest edge to be 15mm. By measuring the photo, I see that the distance between 0 and 1 is 7.5mm and the distance between 27 and 28 is 0.9mm.  Is there a way to get these results from a formula?  It doesn't need to be 100% accurate: it's for photometric software I'm writing, so it's only going to be used as a rough guide. Prove that $\cos\theta$ is root of equation $8x^3-4x^2-4x+1=0$, given $\theta=\frac{\pi}{7}$.   I put $\cos\theta$ in equation, but couldn't show the left-hand side to be zero. how long does the care take to reach the last pole algebra word problem There are 20 poles with a constant distance between each pole. A car takes 22 seconds to reach the 12th pole starting from the first pole, with a constant speed between each pole. how many seconds will it take to reach the last pole?   Is the answer 36 seconds?   since $\frac{22} {12}  = 1.83 $  $1.83 \times 20 = 36.67 $ For some time, Microsoft has supported an option to remove the limit of 260 symbols for NTFS file name length.  The web is full of articles, describing what should be done.   Hit the Windows key, type gpedit.msc and press Enter. Navigate to Local Computer Policy > Computer Configuration > Administrative Templates > System > Filesystem > NTFS. Double click the Enable NTFS long paths option and enable it.   However, I don't have such option. Please see the screenshot below:    Am I missing some Windows update, or is it something else? Subsurf and mirror modifier creating a split in mesh In all the discussions about how the heavy elements in the universe are forged in the guts of stars and especially during a star's death, I usually hear that once the star begins fusing lighter atoms to produce iron (Fe) that's the end of the star's life and the whole system collapses onto itself; and based on how massive the star was initially, it has different outcome - like a white dwarf, a neutron star or a black hole. I have rarely heard a detailed explanation of how the elements heavier than iron are produced. I would appreciate a convincing explanation of this process. How to specify a short int literal without casting? Is the axiom of choice needed in proving that metric spaces in which every infinite subset has a limit point are compact? $X$ and $Y$ are independent rv having pdf $f(t)=\frac{1}{\pi} \frac{1}{1+t^2}$ determine pdf of $Z=\frac{X+Y}{3}$