Prove with the mean value theorem that $x-\frac{x^2}{2}&lt;\ln(1+x)&lt;x$ in $(0,\infty)$  Approach $f(x) := \ln(1+x) $ with the mean value theorem in $[0,x]$  $\frac{1}{1+\xi}= \frac{\ln(1+x)-0}{x-0}$  $\frac{1}{1+\xi}$ takes the biggest value when $\xi$ is $0$  and so $1 &lt;\frac{\ln(1+x)}{x}$ multiply with x and you get $x&lt;\ln(1+x)$  I can't prove the second part.
How can I prove that $$\displaystyle x-\frac {x^2} 2 &lt; \ln(1+x)$$ for any $x&gt;0$  I think it's somehow related to Taylor expansion of natural logarithm, when:  $$\displaystyle \ln(1+x)=\color{red}{x-\frac {x^2}2}+\frac {x^3}3 -\cdots$$  Can you please show me how? Thanks.
Unintentional mouse movements, am I paranoid? Print numbered paragraph after title with Biblatex My question is related with Biber and Biblatex (latest versions). I need to quote the number of a paragraph of an old Christian text which is published within a book printed nowadays.   More clearly, I'm working with this Biblatex item (it's in Italian):  @INBOOK{Clemente1984Lettera,   author = {Clemente di Roma},   title = {Lettera di Clemente Romano ai Corinti},   year = {1984},   booktitle = {I Padri apostolici},   editor = {Quaquarelli, Antonio},   location = {Roma},   pages = {41-92},   publisher = {Città Nuova} }   I quote it in a common LaTex file with the BibTex key:  \cite[42]{Clemente1985Lettera}   With a humanistic Biblatex style, the final citation is:  Clemente di Roma, Lettera di Clemente Romano ai Corinti, in "I Padri apostolici", edited by A. Quaquarelli, Roma 1985, p. 42.  Anyway, I need to quote specifically the n. 2 of the letter of Clement, so I have to print this number just after the title "Lettera di Clemente Romano ai Corinti". So, I'd like to have this citation:  Clemente di Roma, Lettera di Clemente Romano ai Corinti, n. 2, in "I Padri apostolici", edited by A. Quaquarelli, Roma 1985, p. 42.  Now, I can find no Biblatex field to store the "n. 2" text, and no way to print it after the title. Of course, I could create a specific item in the bibliographic file in which the title field, or another one which is supposed to follow it, contains the "n. 2" indication. Anyway, this solution is very bad to me, because I should fill my archive with items of specific citations, and not of single titles.  What I'm looking for is what I've already implemented in the Bibfilex citations system (), which is moreover far less powerful that Biber: the possibility to add a pipe in the postnote of a BibTex key to separate a possible indication of a number of a paragraph, to be printed just after the title, from the pages of the volume. So a key like that  \cite[2|42]{Clemente1985Lettera}   should print   Clemente di Roma, Lettera di Clemente Romano ai Corinti, n. 2, in "I Padri apostolici", edited by A. Quaquarelli, Roma 1985, p. 42.  Anybody knows how to do that with Biber and Biblatex?  Thanks. What is the technical term for when people thinks the actions of the few represent the many.  Like when people thinks everyone in a specific group behaves a certain way cause of how a few people in the group behave, including non stereotypical behaviours. I thought it was "generalisation" but I couldn't find any articles using that term. I've found strange behavior when closing this  - I definitely see that there are 3 closing votes and 2 of them are at "Off-topic" section:   -but there are no explanations on section itself:   -while, normally, it's expected that it will appear. In custom comments section there are also no closing-votes displayed.   Perhaps it's just my browser (which is FF 26 x64 under Win) - so marking this as a "possible bug". Thank you for your attention. I would like to create a md5 checksum list for all files in a directory.  I want to cat filename | md5sum &gt; ouptput.txt. I want to do this in 1 step for all files in my directory.   Any assistance would be great. Can I do it with reflection or something like that? Connected to wifi successfully, but no internet connection How to make /sbin/shutdown, /sbin/reboot etc. require sudo again in 16.04? For whatever reason, we do no longer need to be root (or using sudo) to run /sbin/shutdown, /sbin/reboot etc.  This seems to be because those executables are now symlinks to /bin/systemctl which handles everything as normal user.  However, what if I want shutdown and reboot to require root authentication again? How can I achieve this? Where does "pay through the nose" originate? What are the relationship between administrators groups of domain, dev computer and of Sharepoint services? I know-know I can search and read but anyway I would need to ask in order to escape possible typos, personal misunderstandings and to better remember.     I develop on a machine with installed Sharepoint 2010 Sever where I have local groups:     Administrators Description: Administrators have complete and unrestricted access to the computer/domain   WSS_Admin_WPG Description: Members of this group have write access to system resources used by Microsoft SharePoint Foundation      **Does the description of "Administrators" ("computer/domain") imply all privileges of "WSS_Admin_WPG" ?    Does the description of **WSS_Admin_WPG (in part of "system resources used by Microsoft SharePoint Foundation") imply Sharepoint Server?**    The WSS_Admin_WPG group has entries of:     "BUILTIN\Administrators (S-1-5-32-544)" group   and some user domain accounts   Is "BUILTIN\Administrators (S-1-5-32-544)" group the same (alias to?) as "BUILTIN\Administrators" entry I observe among Sharepoint Farm Administrators (through "Central Administration")?       Is the entries of domain accounts excessive (or necessary)     in "WSS_Admin_WPG" if these accounts make part of "Administrators" group?     in "Farm Administrators" if these accounts are part of "BUILTIN\Administrators (S-1-5-32-544)" group?     What are relationship (difference) between "BUILTIN\Administrators" and "Administrators"? Why is the Apps flash storage so high and how can it be reduced? This is the disk storage info:    This is the Applications folder info:    Why is the storage usage so high and how can it be reduced? This is on a MacBook Pro Retina 13" with OSX 10.9.5.  Edit: I had tried all the suggestions made by @user3439894 in his answer but didn't mention it before posting. Is it possible to detect a virus with taskmanager? If I would have a running virus on my system, would I be able to see the process in taskmanager? I mean, would it be possible for a running virus to circumvent the taskmanager so the process doesn't appear in the tasklist of windows7?  Or in other words. If I really now all the processes in taskmanager to be secure, I also know that my PC is clean? Why didn't Sauron become invisible while wearing the One Ring? I'm curious as to why Sauron is the only being/person shown wearing the One Ring who does not become invisible?   Isildur, Gollum, Bilbo, Frodo, and Samwise all become invisible while wearing the One Ring.  Why didn't Sauron become invisible while wearing the one ring?  I became interested after reading Pearsonartphoto's question asking what effect wearing the One Ring would have had on . NOT IN vs NOT EXISTS Outlier Detection on skewed Distributions Under a classical definition of an outlier as a data point outide the 1.5* IQR from the upper or lower quartile, there is an assumption of a non-skewed distribution. For skewed distributions (Exponential, Poisson, Geometric, etc) is the best way to detect an outlier by analyzing a transform of the original function?  For example, distributions loosely governed by an exponential distribution, could be transformed with a log function - at which point is it acceptable to look for outliers based on the same IQR definition? I 'had' better get going (Why 'had'??) Can I use film that's 20+ years old? I cannot mount Windows 8 partition disk on Ubuntu 12.10.  My Windows 8 is installed on "C" partition, and I have another "D" partition.  I can easily mount "D" partition, see files and make changes on them. Actually I also have a problem with this: when I copy a file from ubuntu to "D", I cannot see this file on Windows 8.  My main problem is when I try to mount "C" partition I get this:     Unable to mount 121 GB Volume   Error mounting /dev/sda3 at /media/furkan/06DEC78FDEC77589: Command-line `mount -t "ntfs" -o "uhelper=udisks2,nodev,nosuid,uid=1000,gid=1000,dmask=0077,fmask=0177" "/dev/sda3" "/media/furkan/06DEC78FDEC77589"' exited with non-zero exit status 14: Windows is hibernated, refused to mount.   Failed to mount '/dev/sda3': Operation not permitted   The NTFS partition is hibernated. Please resume and shutdown Windows   properly, or mount the volume read-only with the 'ro' mount option, or   mount the volume read-write with the 'remove_hiberfile' mount option.   For example type on the command line:           mount -t ntfs-3g -o remove_hiberfile /dev/sda3 /media/furkan/06DEC78FDEC77589 Mirror modifier. Mirrored half is not aligned, but rotated I was working on my character's hand when I realized half of the body was suddenly deleted. I am trying to recreate the other half, but it just won't mirror properly in the center. I tried moving the origin to the 3D cursor while it was completely centered, but the mirror modifier is still applied at an angle. What is the term for a factorial type operation, but with summation instead of products?