Can you help me prove this two sigmas: $$\sum_{k=2}^n 2^{1-k} = 2^{-n}(2^n - 2) $$ and this $$\sum_{k=2}^n k2^{1-k} = 3-2^{1-n}(n+2) $$ Is there any common way to solve this kind of tasks? Thank you very much for help.
I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived.  For example, when r = 2, the formula is given by: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ according to   Thanks!
How do I reset an old iPod? My daughter gave her little brother her iPod Touch 8g. I have been trying to update the software and would like to get it rest to a factory clean state so I can linked it to the iTunes that already syncs to his iPad.   How can I do a clean up and wipe of everything from an iPod? Can Mjolnir be lifted & wielded by anyone if Thor is touching it? In The Big Bang Theory episode , Penny, Amy, and Bernadette are having an argument if Thor's hammer:   Can be lifted by anyone in space.     Can be lifted &amp; wielded by anyone if Thor is touching it.   The first argument is settled in .  But the second argument remains unsolved.  Can Thor's hammer be lifted &amp; wielded by anyone if Thor is touching it? In QGIS, I have been learning a few expressions to style my trail maps but still know little and am stumped on what to use to filter my symbol data. I am showing Parking locations on my map. I use a marker symbol as a background element, so I can do callouts. I created a data field and added values to display beside this symbol a number that references a list of parking addresses in the book's text. All good except if the field has no number to display, it is NULL, it does not display my marker symbol. I need to filter this, so it does show even when there is no value in the field. How can I retrieve data within my widget I have a Widget which I am creating but my quesiton is how do I access my custom table which I created via my plugin. I am attempting to fill the dropdown from the table called tfp which contains the column liters which I wish to popluate the values into the drop down   class my_plugin extends WP_Widget {  // constructor function my_plugin() {         parent::WP_Widget(false, $name = __('Thomsons Fuels Widget', 'wp_widget_plugin') );  }  // widget form creation function form($instance) {  //Check values if( $instance) {  $title = esc_attr($instance['title']);  $dropdown= esc_attr($instance['dropdown']);  $textarea = $instance['textarea'];  } else {  $title = '';  $textarea = ''; } ?&gt;   &lt;p&gt; &lt;label for="&lt;?php echo $this-&gt;get_field_id('dropdown'); ?&gt;"&gt; &lt;?php _e('Liters',   'wp_widget_plugin'); ?&gt;&lt;/label&gt; &lt;select&gt; &lt;option value="volvo"&gt;Volvo&lt;/option&gt; &lt;/select&gt;  &lt;/p&gt;   How I created the table in the plugin   global $wpdb; global $custom_table_example_db_version;  $table_name = $wpdb-&gt;prefix . 'tfp'; // do not forget about tables prefix  // sql to create your table // NOTICE that: // 1. each field MUST be in separate line // 2. There must be two spaces between PRIMARY KEY and its name //    Like this: PRIMARY KEY[space][space](id) // otherwise dbDelta will not work $sql = "CREATE TABLE " . $table_name . " (   id int(11) NOT NULL AUTO_INCREMENT,   liters VARCHAR(100) NOT NULL,   price VARCHAR(100) NOT NULL,      postcodearea varchar(20) NULL,         dft varchar(20) NULL,   PRIMARY KEY  (id) );";  // we do not execute sql directly // we are calling dbDelta which cant migrate database require_once(ABSPATH . 'wp-admin/includes/upgrade.php'); dbDelta($sql);  // save current database version for later use (on upgrade) add_option('custom_table_example_db_version', $custom_table_example_db_version); '+' and 's' in permission strings If the ls -l command gives me a permission string like  rwsr-s--x   What does the 's' mean? The only sources I found mention that it can be present sometimes but do not elaborate.  What does a '+' instead of a '-' mean? I have found mentions of 'extended permission' but nothing clear. We should be able to find questions or answers that we have deleted I deleted one of my questions and would like to undelete it. This requires finding the question.  I think you shouldn't be prevented from seeing your own posts that you have deleted. It should be easier than me remembering the link or checking the browser history to easily look these up.   I would currently need 10K to see my deleted items and even then if I can find their URL. How to inject Spring Bean for factory method requiring MyClass.class parameter Best way to clone an installation (copying to identical hardware) Kind-of like  but slightly different (I think), in that I have 6 identical Acer Aspire Revo R3610 machines. One is (almost) configured to my requirements - when I'm done preparing it I'd like to make the other 5 machines absolutely the same. I'm very new to Ubuntu, what's the most straightforward (easiest) way of doing this?  The machines are going to live on different networks if that might otherwise be a problem (eg with Windows you can clone disks but you then have to make registry changes afterwards if they're going to run on the same network etc). The hardware in all 6 machines is, I stress, the same!  How can I efficiently clone one source image on to these identical machines?  Please restrict one software/solution per answer I would like to either prevent or handle a StackOverflowException that I am getting from a call to the XslCompiledTransform.Transform method within an Xsl Editor I am writing. The problem seems to be that the user can write an Xsl script that is infinitely recursive, and it just blows up on the call to the Transform method.  (That is, the problem is not just the typical programmatic error, which is usually the cause of such an exception.)  Is there a way to detect and/or limit how many recursions are allowed?  Or any other ideas to keep this code from just blowing up on me? How does ZFC describe addition? Surprisingly, the Wikipedia article on addition doesn't contain the answer. I looked  elsewhere online for it, but didn't find it.   Intuitively, the cardinal of the union of two sets seemed appealing. But that approach doesn't work, given that the union of 2 and 2 is 2. I wondered whether ordered sets might be essential to the answer.   So, how does ZFC describe addition? How to make a multiboot USB key? I want to cut my 8 GB USB key into several partitions to use  for a Windows XP (maybe d before).   I also want to put the  remix pack into it such that it has the second bootable OS. I want to tweak and modify it a little because if I can I want to switch Ubuntu 9.04 to the  version.  How to do it easily?  IMP: in short I want to make a dual-boot USB key with Windows XP SP3 and Ubuntu 8.04.3 LTS. In Excel, the window can be frozen in place in such a way that you can always see the name of the column even though you go way down below the 'fold' of the screen.  On SE, when you scroll, the whole page goes up (or down). This means the top navigation bar is scrolled off screen. I'm not sure if it would be a good idea or not to fix the top navigation to the top of the screen and let the rest of the content scroll.  What do you think about this, in terms of pros and cons?  [Update]  With the new Top navigation bar, kudos to the design team, I think this question becomes relevant again. I'm aware of the busload of downvotes, but still.  The top navigation bar doesn't occupy much space and has two notification areas:   Inbox Achievements   Having these two notifications visible at all times would be nice, IMO.  [Update]  I've just noticed that I haven't used the word "optional" in this discussion... My bad. In Excel, the Freeze function is controlled by the user. You can optionally Freeze part of the interface, and you can eliminate the Freeze at will. Why do I keep getting told I posted less than 15 characters when I posted more than 15? I tried to post the following comment and it claims it was less than 15 when it was more.  Based on Tim's comment and testing I believe it would not let me post because with @Shadow removed it is less than 15. However, the characters left message indicates I have passed the 15 character minimum thus it is wrong.  This is the bug I'm reporting:  The characters remaining message and the characters left message do not take into account striping out @name prefixes. I came across this statement while studying electric currents and I am confused:  "There is no electric field inside a conductor. Hence no current can flow through it".  Is there a fallacy in this statement?  Any help would be appreciated.  The conductor being discussed is ideal. What is the driving force for the current in this case? I am having Problems converting user rights. I want to assign the following rights to certain files/folders but dont know hat to translate them into a format which chmod will work with like chmod 755.  -rwxrwxrwx    will be?  And  drwxr-xr-x   will be?  And finally  -rw-r--r--   will be? Why does Python pep-8 strongly recommend spaces over tabs for indentation? Is it possible to instantiate an object at runtime if I only have the DLL name and the class name, without adding a reference to the assembly in the project? The class implements a interface, so once I instantiate the class, I will then cast it to the interface.  Assembly name:     library.dll    Type name:      Company.Project.Classname     EDIT: I dont have the absolute path of the DLL, so Assembly.LoadFile won't work. The DLL might be in the application root, system32, or even loaded in the GAC. $X$ and $Y$ are independent $U(0,1)$ random variables.   I have to find $E[X|X&gt;Y]$.   I found it out by integrating $\int_0^1 \int_y^1 x dxdy$ to get $\frac{1}{3}$ as the answer but this is wrong according to the answer key. What is the mistake that I have made? I am running a fresh install of Linux Mint Nadia (14). I am following the instructions on Vagrant  but have gotten stuck on the . It seems the Vagrant box cannot connect outside and so I can't install anything using either Chef or Puppet.  In the basic Vagrant resolve.conf contains nameserver 10.0.2.3. But with that set I can't ping us.archive.ubuntu.com.  If I change it to 8.8.8.8 then I can ping us.archive.ubuntu.com but it does not stay set, and after a reboot it changes back to 10.0.2.3 - so provisioning fails again.  Ideally I would like for 10.0.2.3 to work on my setup. Failing that I would like a way to permanently change resolv.conf so that I can do provisioning. I built a ZI model and it is producing predicted values that are from a very small range when compared to the observed values. Plus it does not produce any zeros. See the fitted vs. observed graph below.     My model structure is given below. My response is a count variable and the predictor is continuous water level measurements (scaled).   m.u.meanwl.gu &lt;- zeroinfl(GU ~ MeanWLScaled | MeanWLScaled, data = uluabat_wl_fg, dist = "negbin", link = "logit")   I need help to figure out what's wrong and fix it. Thanks in advance.