is the sequence $\{a_n\}$ is strictly increasing or strictly decreasing?
Monotonicity of the sequences $\left(1+\frac1n\right)^n$, $\left(1-\frac1n\right)^n$ and $\left(1+\frac1n\right)^{n+1}$
I want to buy a new notebook in the near future to play games on it. I'm looking forward for the new Steam for Linux client. My question is that where does the Steam install the games, on the home partition, or in the usr folder, or somewhere else? I do not use pre-release (i.e. beta) versions of either Steam or Ubuntu. Example to prove that $ C^1[0,1] $ is not a Banach space for the uniform norm? The space $ C^1[0,1] $- the space of all continuously differentiable functions on $ [0,1]$ is not a Banach space with respect to the sup norm,$ \|.\|_{\infty} $ since the uniform limit of a continuously differentiable function need not be differentiable.  How can I illustate this statement using a counter example? Can I use $ f_{n}=\frac1 n \sin nx $ as a counter example?  Also, is $ C^1[0,1] $ is same as the space given by $ X=\{f\in C^1[0,1]:f(0)=0 \}$. Can I use the same example to show that this is not a Banach space?  More help is appreciated! Thanks! How do I use SSMS to compare Database 1 Table 1 against Database 2 Table 1? I have two databases - Database1 and Database2.  Both databases contain a table that has similar structure, exemplified as follows:   =========================================================================================== | ID | Name | PhoneNoFormat | DialingCountryCode | InternationalDialingCode | InternetTLD | =========================================================================================== |    |      |               |                    |                          |             | ===========================================================================================   However, due to some reason, (a) one of the tables in one of the databases has data that is not exactly the same as (b) that contained in the other table in the another database.  So, how can I compare Database1.Table1 against Database2.Table1?  I tried using the following query, but nothing happened so was wondering if I have to rewrite it:  SELECT MIN(TableName) as TableName,        ID,        Name,        PhoneNoFormat,        DialingCountryCode,        InternationalDialingCode,        InternetTLD  FROM  (    SELECT  'Table A' as TableName,           A.ID,           A.Name,           A.PhoneNoFormat,           A.DialingCountryCode,           A.InternationalDialingCode,           A.InternetTLD    FROM [D:\DATABASE1.MDF].[dbo].[Table1] AS A    UNION ALL    SELECT 'Table B' as TableName,          B.ID, B.Name,          B.PhoneNoFormat,          B.DialingCountryCode,          B.InternationalDialingCode,          B.InternetTLD    FROM [D:\DATABASE2.MDF].[dbo].[Table1] AS B  ) tmp  GROUP BY ID, Name, PhoneNoFormat, DialingCountryCode, InternationalDialingCode, InternetTLD  HAVING COUNT(*) = 1  ORDER BY ID Wifi disabled, no hardware switch available, rfkill list reads "Hard Blocked" anyway on an Acer Aspire One Can I use an nVidia Geforce 9600GT for triple-monitor? Possible Duplicate:          My work PC has (as you can guess from the title) a single Geforce 9600GT card. According to the  for this card, both DVI ports are Dual-Link DVI. Both are currently in use on Dell 19" 1280x1024 screens. I'm running Windows 7, with Aero enabled, for what that's worth.  What I'm wondering is if I can use a standard DVI splitter cable to drive a third screen from one of the ports -- specifically a Samsung 22" 1680x1050. I'm really outgrowing the two-screen setup, and am strongly considering bringing in my home monitor (which is almost never used) to bolster my usable space. If I can't, then my alternative is to replace one of these Dells with that Samsung, to give me at least a "little" more space.  (Before you ask, there's probably no chance I can get my company to spring for a second video card and third monitor. Even if I could, it would not be politically expedient, as a number of my other coworkers are already pretty jealous at my setup.) Can I use set instead of List to resolve the duplicate id issue If you take a look into this issue     I have provide an answer of using set&lt;Sobject&gt; instead using list&lt;Sobject&gt;  i.e.  if( !contacts.values().isEmpty() ){   set&lt;Contact&gt; setContacts = new set&lt;Contact&gt;();   for( Contact objCon : contacts.values() ){     if(!setContacts.contains(objCon)){         setContacts.add( objCon );     }   }  update new list&lt;Contact&gt;(setContacts); }   even if(!setContacts.contains(objCon)) is not needed  if( !contacts.values().isEmpty() ){   set&lt;Contact&gt; setContacts = new set&lt;Contact&gt;();   for( Contact objCon : contacts.values() ){         setContacts.add( objCon );   }  update new list&lt;Contact&gt;(setContacts); }   So is there any condition/use case where this answer might fail? Why apt can't find the source code of any package? How can I check the actual size used in an NTFS directory with many hardlinks? On a Win7 NTFS volume, I'm using cwrsync which supports --link-dest correctly to create "snapshot" type backups. So I have:  z:\backups\2010-11-28\cygdrive\c\Users\... z:\backups\2010-12-02\cygdrive\c\Users\...   The content of 2010-12-02 is mostly hardlinks back to files in the 2010-11-28 directory, but there are a few new or changed files only in 2010-12-02. On linux, the 'du' utility will tell me the actual size taken by each incremental snapshot. On Windows, explorer and du under cygwin are both fooled by hardlinks and shows 2010-12-02 taking up a little more space than 2010-11-28.   Is there a Windows utility that will show the correct space acutally used? Rkhunter warnings help I installed Ubuntu 14.04 a few days ago. I've made a scan with a RkHunter and got a few warnings that I don't really know what to do with. I read about false positives a little bit but I didn't find the answer for myself really. I got those errors:  Checking /dev for suspicious file types         [ Warning ] [19:44:16] Warning: Suspicious file types found in /dev: [19:44:16]          /dev/.udev/rules.d/root.rules: ASCII text [19:44:16]   Checking for hidden files and directories       [ Warning ] [19:44:16] Warning: Hidden directory found: /etc/.java: directory [19:44:16] Warning: Hidden directory found: /dev/.udev: directory [19:44:16] Warning: Hidden file found: /dev/.initramfs: symbolic link to `/run/initramfs' Checking for group file changes                 [ Warning ] [19:44:16] Warning: Group 'winbindd_priv' has been added to the group file.   Should I be worried ? Ubuntu with spyware? Richard Stallman states that Ubuntu "has installed surveillance code". See:   Is this true?   I agree with him that "Any excuse Canonical offers is inadequate....". gedit: search and replace with regular expressions I want to replace this with gedit:   1299465 |  2003415 | 2015-09-06 05:35:34.59662+02  1299449 |  2009400 | 2015-09-06 05:35:32.301683+02  1299450 |  2008465 | 2015-09-06 05:35:32.451393+02  1299457 |  2015211 | 2015-09-06 05:35:33.451049+02   to that:  2003415 2009400 2008465 2015211   How can I do this with gedit?  Background: I want to do it with gedit and not sed/python/.... since the replace needs to be done interactive. Problems with booting the system I have yesterday updated packages and today the system cannot boot properly. After seeing the Kubuntu logo there is a black screen which shows repeatedly.     Stopped User Manager for UID 121.      Removed slice User Slice of gdm.      Created slice User Slice of gdm.       Starting User Manager for UID 121...      Started Session c3 of user gdm.       Started User Manager for UID 121.   Please see the screenshot below This question is from [Number Theory George E. Andrews 1-1 #3].     Prove that $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1}).$$   This problem is driving me crazy. $$x^n-y^n = (x-y)(x^{n-1}+x^{n-2}y+\dots +xy^{n-2}+y^{n-1)}$$  $(x^n-y^n)/(x-y) =$ the sum for the first $n$ numbers and then I added $(xy^{(n+1)-2}+y^{(n+1)-1})$ which should equal $(x^{n+1}-y^{n+1})/(x-y)$ but I can't figure it out    This is a similar problem in the book and I tried this method but it wasn't working out      $\quad$Thereom $\bf1$-$\bf2$: $\,\,\,\,$ If $\,x$ is any real number other than $1$, then $$\sum_{j=0}^{n-1}x^j=1+x+x^2+\ldots+x^{n-1}=\dfrac{x^n-1}{x-1}.$$    $\quad$Remark: $\displaystyle\sum_{j=0}^{n-1}A_j$ is shorthand for $A_0+A_1+A_2+\ldots+A_{n-1}.$     $\quad$Proof: Again we proceed by mathematical induction. If $n=1$ then $\displaystyle\sum_{j=0}^{1-1}x^j=x^0=1$ and $(x-1)/(x-1)=1$. Thus the theorem is true for $n=1$.  $\quad$ Assuming that $\displaystyle\sum_{j=0}^{k-1}x^j=(x^k-1)/(x-1)$, we find that $$ \eqalign{ \sum^{(k+1)-1}_ {j=0}x^j &amp; = \sum^{k-1}_ {j=0}x^j+x^k=\dfrac{x^k-1}{x-1}+x^k \\ &amp;= \dfrac{x^k-1+x^{k+1}-x^k}{x-1}\\ &amp;= \dfrac{x^{k+1}-1}{x-1}. }$$   Hence condition $(\rm ii)$ is fulfilled, and we have established the theorem.    $\quad$Corollary $\bf1$-$\bf1$: $\,\,$ If $\,m$ and $n$ are positive integers and if $m&gt;1$, then $n&lt;m^n.$ What is the difference between fixed effect, random effect and mixed effect models? In simple terms, how would you explain (perhaps with simple examples) the difference between fixed effect, random effect and mixed effect models? When I tried to connect my Ubuntu 13.10 to my VM Centos 6.5 using ssh appear the message "Connection refused" or "No route to host" even than I change the port, noting change.    I need to use SSH, VPN and Samba, but this IP in my VM Centos do not work.  How Can I resolve this problem? Windows 10 April 2018 Update added an extra Language and I cannot remove it I have just updated to Windows 10 April 2018 Update (1803 / 17134.1)  I had my laptop configured completely as UK with only a UK keyboard layout  Since the update, a US keyboard has been added so I now see the switch icon in my task bar    It appears that all the keyboard language settings are now migrated to the new Settings app and no longer in the old Control Panel    However, as can be seen from the screenshot above, the Remove button under the newly added US layout is greyed out (as is the UK one)  How can I remove the US layout?  UPDATE 1: I just updated my desktop and it hasn't added a US keyboard!  UPDATE 2: I added a third language (French) and was able to remove it, the button was enabled. The US remove button remained disabled both after adding French and after removing it.  UPDATE 3: I added another English language (Australian) and this cannot be removed! Button is disabled  UPDATE 4: I followed this  except I removed the values from the keys for all users and also the values under the Substitutions key. The task bar switch icon is gone but I still have all 3 languages in Settings still Prove QM-AM inequality $$\dfrac{x_1^2+ x_2^2 + \cdots + x_n^2}n \geq \left(\dfrac{x_1+x_2+\cdots+x_n}n\right)^2$$   I don't think AM, GM can be used here. And simple expansion doesn't help too. What should I do? I am trying to understand the sentence  the mean minimizes the mean squared error.   from wikipedia .  From a previous post  I can see a formal proof which according to the authors prove that the mean minimaze. But my objection on that prove is that is you substitute m with median all the reasoning still stands.  Therefore, I restricted myself to one dimension and built the following example :   MSE = 1/n[SUM(Y_hat_i - Y_i)**2]   Given the following distribution Y = [5,3,2,7,4], minimizing MSE I intended as in the following: I should find a function Y_hat such that when it is applied to the MSE formula (more specifically to each element of the Y distribution), we are sure that we obtain the minimum value. For simplicity, I could consider the Y_hat as any measure or function for the central tendency, including, mean, median, mode, etc.  Lets suppose, Y_hat = mean (=4.2 for our distribution), therefore, I did all the calculations such as:  1/5 [(4.2-5)**2 + (4.2-3)**2+ (4.2-2)**2 + (4.2-7)**2  + (4.2-4)**2 ] = 2.96   Let's suppose, Y_hat = median (=4 for our distribution), therefore, I did all the calculations such as:  1/5 [(4-5)**2 + (4-3)**2+ (4-2)**2 + (4-7)**2  + (4-4)**2 ] = 2.8   If I did all the calculation correctly, the result shows that for this case, the median is minimizing better than the mean.  I am sure, I am doing something wrong and missing important aspects of the overall reasoning.  Please, could you provide any clarification on the above topic? I would really appreciate it.  Many Thanks in advance, Best Regards, Carlo Navier-Stokes system After update, my Thunderbird stopped getting minimized with the "MinimizeToTray revived" add-on. Is there any way to minimize my Thunderbird? Any other add-on that works with the new version?