There is the numerical series of $\sqrt{n^{2} +1} -n$. I tried to use D'Alembert criterion to conclude if it converges but the limit is equal to 1. Even the Cauchy criterion does not work. Thank you for your guidance.
Test the series whose general term is: $$\sqrt{n^2 + 1} - n$$   This is from Higher Algebra by Henry Sinclair. I'm struggling to understand how this series is divergent. If you do the following expansion: $$\sqrt{n^2 + 1} - n$$ $$= n\biggl(1 + \frac{1}{n^2}\biggr)^\frac{1}{2} - n$$ $$= n + \frac{1}{2n} - \frac{1}{8n^3} + ... - n$$ $$= \frac{1}{2n} - \frac{1}{8n^3} + ... $$  The remaining series appears to converge to zero as $n$ approaches infinity.  Their explanation is similar to mine, but at the end they say that the above series approaches $\frac{1}{2n}$. From there they directly say that the series is divergent. Earlier in the chapter, it's proved that the expansion of $(1 + x)^n$ is convergent for $x &lt; 1$. In this case, $\frac{1}{n^2} &lt; 1$ as $n$ approaches infinity, so I would assume that the series converges.  The solution to the problem is given  (number 17)  Any help is appreciated. Thanks.
Ruby block and unparenthesized arguments I extracted simple example:  require 'pp' x = 1..3 pp x.map do |i| {:value =&gt; i, :double =&gt; (i*2)} end pp x.map { |i| {:value =&gt; i, :double =&gt; (i*2)} }  pp(x.map do |i| {:value =&gt; i, :double =&gt; (i*2)} end) pp(x.map { |i| {:value =&gt; i, :double =&gt; (i*2)} })   I am wondering why first pp produces:  [1, 2, 3]   While all the oders are giving:  [{:value=&gt;1, :double=&gt;2}, {:value=&gt;2, :double=&gt;4}, {:value=&gt;3, :double=&gt;6}]   I assume it has something to do with operator precedence. Where can I find good explanation? Prime factorizations of consecutive numbers, can they both contains powers of greater than or equal to $1000$? The new 17.10 only has 64 bit ISOs. However 32 bit is still supported, there just are no live ISOs for it.  I have a 32 bit machine with WIFI, and I need to install from a USB disk. I tried the "mini.iso" but it doesn't have WIFI support.  What options do I have?  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I've tried the several options:   Disable double sided lighting Enable VBO    I've so also reduced the topology of hidden faces. But that´s how far I've reached.  I'm using a late Imac Intel I5 Nvidia GPU GTX 680mx  any tips out there? Adding script conditionally I would like to reload an &lt;iframe&gt; using JavaScript. The best way I found until now was set the iframe’s src attribute to itself, but this isn’t very clean. Any ideas? skype 4.3 cannot connect to account on ubuntu 14.04 LTS I have being using Skype for a long time on my Ubuntu 14.04 LTS. From this morning on, Skype simple cannot connect to my account anymore.  That is my Skype version:     Skype 4.3.0.37   © 2014 Skype and/or Microsoft   My Ubuntu system:  LSB Version:    core-2.0-amd64:core-2.0-noarch:core-3.0-amd64:core-3.0-noarch:core-3.1-amd64:core-3.1-noarch:core-3.2-amd64:core-3.2-noarch:core-4.0-amd64:core-4.0-noarch:core-4.1-amd64:core-4.1-noarch:cxx-3.0-amd64:cxx-3.0-noarch:cxx-3.1-amd64:cxx-3.1-noarch:cxx-3.2-amd64:cxx-3.2-noarch:cxx-4.0-amd64:cxx-4.0-noarch:cxx-4.1-amd64:cxx-4.1-noarch:desktop-3.1-amd64:desktop-3.1-noarch:desktop-3.2-amd64:desktop-3.2-noarch:desktop-4.0-amd64:desktop-4.0-noarch:desktop-4.1-amd64:desktop-4.1-noarch:graphics-2.0-amd64:graphics-2.0-noarch:graphics-3.0-amd64:graphics-3.0-noarch:graphics-3.1-amd64:graphics-3.1-noarch:graphics-3.2-amd64:graphics-3.2-noarch:graphics-4.0-amd64:graphics-4.0-noarch:graphics-4.1-amd64:graphics-4.1-noarch:languages-3.2-amd64:languages-3.2-noarch:languages-4.0-amd64:languages-4.0-noarch:languages-4.1-amd64:languages-4.1-noarch:multimedia-3.2-amd64:multimedia-3.2-noarch:multimedia-4.0-amd64:multimedia-4.0-noarch:multimedia-4.1-amd64:multimedia-4.1-noarch:printing-3.2-amd64:printing-3.2-noarch:printing-4.0-amd64:printing-4.0-noarch:printing-4.1-amd64:printing-4.1-noarch:qt4-3.1-amd64:qt4-3.1-noarch:security-4.0-amd64:security-4.0-noarch:security-4.1-amd64:security-4.1-noarch Distributor ID: Ubuntu Description:    Ubuntu 14.04.3 LTS Release:    14.04 Codename:   trusty   Any suggestion why or how to fix it?  Thanks in advance. How to move a visible file to a hidden folder? I've only been using Ubuntu for two weeks and have been able to find answers to any problem I had, however this one defies my searches.  I want to move a visible programme file (wallch) to a hidden directory ./config/autostart.  I have been able to enter this directory and the subject file is not there and I want to have this program start at bootup.  Can anyone assist?  Thanking you How to change phone number in Google account? I gave Google my phone number to verify it, not knowing that it would change it, and now my phone number is completely different? What have I done, and how do I unverify it and change my phone number back? I've had  on the web since 1999, and I'd like to add a page on why complex numbers are (or should be) considered to be numbers.  I'm frequently asked that question.  This is not a question about the existence or usefulness of complex numbers as they do exist as mathematical entities and they are very useful.  It is a question about what makes them numbers.  It could be broadened to ask what constitutes numbers, but I'm specifically interested in complex numbers.  Perhaps it's just historical and sociological, but I'm hoping for an answer that there's something intrinsic about complex numbers that makes them fit into the category of number. I have bumped into an issue that doesn't allow me to do anything related to apt-get: install, remove, etc. Whenever I try to do something (e.g. sudo apt-get install firefox -f) it gives me the same error message:  Reading database ... dpkg: unrecoverable fatal error, aborting:  unable to open files list file for package `fontconfig-config': Input/output error E: Sub-process /usr/bin/dpkg returned an error code (2)   I have tried to deal with the package fontconfig-config without success.  I have found that the "Input/output error" is usually linked with physical problems of the hard drive, but I do not think that's the case I am using that hd without any other problem.  I have tried removing the mention to the package in /var/lib/dpkg/status as mentioned .  I have tried deleting all files related to the package in /var/lib/dpkg/info as I found somewhere.  But I still cannot do anything. The funny bit comes when I look for the file that is giving me troubles:   mecho@Ansible-MS-7680:/var/lib/dpkg/info$ ls fontconfig* ls: cannot access fontconfig-config.list: Input/output error fontconfig.list     fontconfig.postinst  fontconfig.preinst  fontconfig.triggers fontconfig.md5sums  fontconfig.postrm    fontconfig.prerm   This is done after I deleted all files ... it looks like fontconfig-config.list still exists but it doesn't show up!  Any idea about how to solve the problem?  I am on kubuntu precise, fontconfig-config_2.8.0-3ubuntu9.1    I do not have smartmontools installed nor any other programme I could use to check the hard disk. The most similar think I seem to have is KDE Partition Manager but it doesn't allow me to check anything (the option appears unavailable). However, I think the problem is not the hd.  How do you suggests me to do the fs check? fcheck is not installed and I cannot think of any other command in bash (as I do not think I have anything else installed that could perform a check).  Thank you.  Edit: I have found the command (silly me), fsck. This is what I got:  /dev/sda1: recovering journal Clearing orphaned inode 4456702 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456700 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456699 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456697 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456694 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456693 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456688 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456481 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456692 (uid=0, gid=0, mode=0100600, size=65536) Clearing orphaned inode 4456689 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456709 (uid=1000, gid=1000, mode=0100600, size=714046) Clearing orphaned inode 4456469 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 3672644 (uid=0, gid=0, mode=0100644, size=1595) Clearing orphaned inode 1324021 (uid=0, gid=0, mode=0100640, size=1980) Clearing orphaned inode 1324014 (uid=0, gid=0, mode=0100640, size=126) Clearing orphaned inode 4456499 (uid=1000, gid=1000, mode=0100600, size=0) Clearing orphaned inode 4456483 (uid=1000, gid=1000, mode=0100600, size=65536) Setting free inodes count to 7147142 (was 7147702) Setting free blocks count to 23891195 (was 23781617) /dev/sda1: clean, 151930/7299072 files, 5292549/29183744 blocks Infinite series $\sum _{n=2}^{\infty } \frac{1}{n \log (n)}$ How to configure a firewall that only allows a few webpages and dropbox and U1 services? A simple regex search and replacement in php for minifying/compressing javascript?