Consider an injective function $f: \Bbb N \rightarrow  \Bbb N$  Show that the series given by:  $$\sum_{n=1}^{\infty } \frac{f(n)}{n^2}$$  diverges.   I'm really not sure how to use the injectivity hypothesis, so any suggestion or any answer would be well received.
We know that $\displaystyle\zeta(2)=\sum\limits_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ and it converges.        Does there exists a bijective map $f:\mathbb{N} \to \mathbb{N}$ such that the sum $$\sum\limits_{n=1}^{\infty} \frac{f(n)}{n^2}$$ converges.      If our $s=2$ was not fixed, then can we have a function such that $\displaystyle \zeta(s)=\sum\limits_{n=1}^{\infty} \frac{f(n)}{n^s}$ converges
I can't get grub menu to show up during boot After trying (and failing) to install better ATI drivers in 11.10, I've somehow lost my grub menu at boot time.  The screen does change to the familiar purple colour, but instead of a list of boot options it's just blank solid colour, and then disappears quickly and boots into the default entry normally.    How can I get the bootloader back?  I've tried sudo update-grub and also various different combinations of resolutions and colour depths in startupmanager application with no success (640x480, 1024x768, 1600x1200, 16 bits, 8 bits, 10 second delay, 7 second delay, 2 second delay...)   edit:   I have already tried holding down Shift during bootup and it does not seem to change the behaviour.  I get the message "GRUB Loading" in the terminal, but then the place where the grub menu normally appears I get a solid blank magenta screen for a while.  Here are the contents of /etc/default/grub  # If you change this file, run 'update-grub' afterwards to update # /boot/grub/grub.cfg. # For full documentation of the options in this file, see: #   info -f grub -n 'Simple configuration'  GRUB_DEFAULT=0 GRUB_HIDDEN_TIMEOUT=0 GRUB_HIDDEN_TIMEOUT_QUIET=true GRUB_TIMEOUT=10 GRUB_DISTRIBUTOR=`lsb_release -i -s 2&gt; /dev/null || echo Debian` GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" GRUB_CMDLINE_LINUX=" vga=798 splash"  # Uncomment to enable BadRAM filtering, modify to suit your needs # This works with Linux (no patch required) and with any kernel that obtains # the memory map information from GRUB (GNU Mach, kernel of FreeBSD ...) #GRUB_BADRAM="0x01234567,0xfefefefe,0x89abcdef,0xefefefef"  # Uncomment to disable graphical terminal (grub-pc only) #GRUB_TERMINAL=console  # The resolution used on graphical terminal # note that you can use only modes which your graphic card supports via VBE # you can see them in real GRUB with the command `vbeinfo' #GRUB_GFXMODE=640x480  # Uncomment if you don't want GRUB to pass "root=UUID=xxx" parameter to Linux #GRUB_DISABLE_LINUX_UUID=true  # Uncomment to disable generation of recovery mode menu entries #GRUB_DISABLE_RECOVERY="true"  # Uncomment to get a beep at grub start #GRUB_INIT_TUNE="480 440 1" I have a Google Apps email (me@business.com) set to auto-forward all mail to my personal Gmail (me@gmail.com).  I have added the me@business.com address to my personal Gmail so I can send mail as me@business.com from within my personal email account.  It's a fairly common way to consolidate email accounts.  Here's my problem.  I want me@gmail.com to filter all emails being forwarded from me@business.com and apply the Business label.  So I tried it two different ways:  A) Use an Alias:   me@business.com forward to me+business@gmail.com Create filter "To:me+business@gmail.com" in me@gmail account Have filter apply Business label   B) Use Googlemail.com address:   me@business.com forward to me@googlemail.com Create filter "To:me@googlemail.com" in me@gmail account Have filter apply Business label   However, neither of these filters work.  I've tested it using a third email address, me@yahoo.com.  Whenever I send an email from me@yahoo.com to me@business.com, the email is properly forwarded to me@gmail.com, and I can reply to me@yahoo.com using the me@business.com address.  But Gmail does NOT apply the Business label.  This is case using either filtering method A or B.  But if I send an email from me@yahoo.com DIRECTLY to me+business@gmail.com or me@googlemail.com, the filter works properly and applies the label.  So there must be something going on with the forwarding of me@business.com to me@gmail.com.  Does anyone know how to get this working? Correlations with unordered categorical variables How to delete a file with a path too long to be deleted I was working with some computer vision source code I found and didn't realize that it creates a cache file that has a VERY LONG name and now I can't delete the 2 of them. I tried to rename it to a shorter name but I can't do anything to the file, I also tried to restart my computer and see if it would just go away. The error message says:  Destination Path Too Long. The file name(s) would be too long for the destination folder. You can shorten file name and try again, or try a location that has a shorter path.   How can I delete the files in this case? Which graphics cards are supported by the new AMDGPU driver in Ubuntu 16.04? What is the most "pythonic" way to iterate over a list in chunks? How can I stop fish from sticking to a stainless steel pan? I think Anne Burrell lied to me. She said fish will always stick to a stainless pan. This is true. No matter how well I lubricate my pan, my fish will always adhere like cement. She also said that the pan will release the fish "when it's ready." Maybe meaning it will eventually become unstuck when the fish is done cooking? Well, I've tried that, and it never works. No matter how long I wait, it never becomes unstuck. Most recently I gave it as much time as I could. When I finally chiseled it off the pan, tearing it apart in the process, I saw that it was starting to burn on that side. I've started going back to non-stick until I get this figured out.  Are there any tips? Am I heating the pan wrong? Does it help to slide the fish around a bit after putting it in the pan? Argument in "thebibliography" I was reading the explanation about  and was totally confused:     The thebibliography environment has an argument which should be a string at least the length of the widest entry label in the source list. Instead of using the numbers generated by the environment as entry labels, you can specify your own by using an optional argument to \bibitem, but remember to increase the length of the argument to the thebibliography command.    What does "widest entry label" mean? Practically, what should I put as the argument in   \begin{thebibliography}{?}   or is the argument not necessary? What is the inverse function of $e^x +x$? 2D random walk within a bounded area Rules of thumb for divisibility Given a number presented in decimal form $n=\sum_{k=0} d_k10^k$ there are some nice thumb rules to decide whether it is divisible with a prime number  $p\in\{7,11,13,17,19,23,29\}$.   For each of this numbers there is a number  $a_p$ such that $p|n\iff p|(a_p\cdot d_0+\sum_{k=0} d_{k+1}10^k)$.  $a_7=-2,\;a_{11}=-1,\;a_{13}=4,\;a_{17}=-5,\;a_{19}=2,\;a_{23}=7,\;a_{29}=3$  E.g. $\;19\cdot 17=323$ and $32+2\cdot 3=38=2\cdot 19$ and $32-5\cdot 3=17$.  Is there a number $a_{31}$ that works the same way? Or $a_p,\;p\ge37$? Elevated Priviliges for Startup Applications in Windows? Is there any way to configure application that requires administrator privilages (elevation) so it's automatically started with elevated privileges and does not request elevation on every startup?  For example, I have Riva Tuner that always starts with Windows. And it always asks for elevation via UAC. I need Riva Tuner to silently start with elevated privileges and without any message from user account control (without disabling it). "E:Read error - read (5: Input/output error), E:The package lists or status file could not be parsed or opened." So I keep trying to install things on my laptop and whenever I try to use sudo apt-get I get the same errors every  time,  SystemError: E:Read error - read (5: Input/output error), E:The package lists or status file could not be parsed or opened.   Also whenever I try to open software Center through my desktop Icon it will just freeze leaving me to force quit.  If I try to use the command software-center in my terminal I get the longest list in the world which is,   che.open() Traceback (most recent call last):   File "/usr/lib/python2.7/dist-packages/gi/overrides/GLib.py", line 629, in &lt;lambda&gt;     return (lambda data: callback(*data), user_data)   File "/usr/share/software-center/softwarecenter/db/pkginfo_impl/aptcache.py", line 261, in open     self._cache = apt.Cache(progress)   File "/usr/lib/python2.7/dist-packages/apt/cache.py", line 105, in __init__     self.open(progress)   File "/usr/lib/python2.7/dist-packages/apt/cache.py", line 150, in open     self._cache = apt_pkg.Cache(progress) SystemError: E:Read error - read (5: Input/output error), E:The package lists or status file could not be parsed or opened. 2013-10-13 10:13:52,685 - softwarecenter.db.pkginfo_impl.aptcache - INFO - aptcache.open() Traceback (most recent call last):   File "/usr/lib/python2.7/dist-packages/gi/overrides/GLib.py", line 629, in &lt;lambda&gt;     return (lambda data: callback(*data), user_data)   File "/usr/share/software-center/softwarecenter/db/pkginfo_impl/aptcache.py", line 261, in open     self._cache = apt.Cache(progress)   File "/usr/lib/python2.7/dist-packages/apt/cache.py", line 105, in __init__     self.open(progress)   File "/usr/lib/python2.7/dist-packages/apt/cache.py", line 150, in open     self._cache = apt_pkg.Cache(progress) SystemError: E:Read error - read (5: Input/output error), E:The package lists or status file could not be parsed or opened. 2013-10-13 10:13:57,365 - softwarecenter.db.pkginfo_impl.aptcache - INFO - aptcache.open() Traceback (most recent call last):   File "/usr/lib/python2.7/dist-packages/gi/overrides/GLib.py", line 629, in &lt;lambda&gt;     return (lambda data: callback(*data), user_data)   File "/usr/share/software-center/softwarecenter/db/pkginfo_impl/aptcache.py", line 261, in open     self._cache = apt.Cache(progress)   File "/usr/lib/python2.7/dist-packages/apt/cache.py", line 105, in __init__     self.open(progress)   File "/usr/lib/python2.7/dist-packages/apt/cache.py", line 150, in open     self._cache = apt_pkg.Cache(progress) SystemError: E:Read error - read (5: Input/output error), E:The package lists or status file could not be parsed or opened.   That's only about one-tenth of the list which is too long to read. I need help, if anyone has an answer please respond "Has been" vs "was" Would you please explain the difference between these two sentences?     Spain ceded (or gave) Puerto Rico to the United States. Since then, Puerto Rico has been a U.S. territory.      Spain ceded(or gave) Puerto Rico to the United States. Since then, Puerto Rico was a U.S. territory.   I am stuck between 'has been' and 'was' in the above sentences. Can the editor for editing tag descriptions get updated to match the normal questions/answers editor? Limit of $n$th root of $n$! I am asked to determine if a series converges or not:  $$\displaystyle\sum\limits_{n=1}^{\infty} \frac{(2^n)n!}{(n^n)}$$  So I'm using the $n$th root test and came up with $\lim_{n \to {\infty}}\frac{2}{n}\times(\sqrt[n]{n!})$ I know that the limit of $\frac{2}{n}$ goes to $0$ when $n$ goes to infinity but what about the $(\sqrt[n]{n!})$? Sudoers blacklist I am making an Android game that is only in portrait screen orientation.  It works great when I run it on my phone, but when I run it on a tablet, even though the screen size is bigger, all of the bitmaps are the same size. Is there any way to make the bitmaps stay the same proportion even on different size screens? Is it possible to win in Don't Starve? I would like to know if it's possible to win in Don't Starve? Or is it just the matter of how long until you die? How can I see the source of built-in JavaScript functions?