How to decide whether I should use “we” in an essay?
Choice of personal pronoun in single-author papers
In this  to a question about a series, a theorem was stated:     If $A= \{a_i \}$ is a set such that $\sum_{i = 1}^{\infty} \frac{1}{a_i}$ converges, then $d(A) = 0$, where $d(A)$ is the  of the set.   My background in number theory is basically zero and all my attempts to prove this have been utterly unseccessful; would anyone please help me, or at least provide me with a hint to prove it? How can I retain HTML form field values in JSP after submitting form to Servlet? How to calculate: $\sum_{n=1}^{\infty} n a^n$ I've tried to calculate this sum:  $$\sum_{n=1}^{\infty} n a^n$$  The point of this is to try to work out the "mean" term in an exponentially decaying average.  I've done the following:  $$\text{let }x = \sum_{n=1}^{\infty} n a^n$$ $$x = a + a \sum_{n=1}^{\infty} (n+1) a^n$$ $$x = a + a (\sum_{n=1}^{\infty} n a^n + \sum_{n=1}^{\infty} a^n)$$ $$x = a + a (x + \sum_{n=1}^{\infty} a^n)$$ $$x = a + ax + a\sum_{n=1}^{\infty} a^n$$ $$(1-a)x = a + a\sum_{n=1}^{\infty} a^n$$  Lets try to work out the $\sum_{n=1}^{\infty} a^n$ part:  $$let y = \sum_{n=1}^{\infty} a^n$$ $$y = a + a \sum_{n=1}^{\infty} a^n$$ $$y = a + ay$$ $$y - ay = a$$ $$y(1-a) = a$$ $$y = a/(1-a)$$  Substitute y back in:  $$(1-a)x = a + a*(a/(1-a))$$ $$(1-a)^2 x = a(1-a) + a^2$$ $$(1-a)^2 x = a - a^2 + a^2$$ $$(1-a)^2 x = a$$ $$x = a/(1-a)^2$$  Is this right, and if so is there a shorter way?  Edit:  To actually calculate the "mean" term of a exponential moving average we need to keep in mind that terms are weighted at the level of $(1-a)$. i.e. for $a=1$ there is no decay, for $a=0$ only the most recent term counts.  So the above result we need to multiply by $(1-a)$ to get the result:  Exponential moving average "mean term" = $a/(1-a)$  This gives the results, for $a=0$, the mean term is the "0th term" (none other are used) whereas for $a=0.5$ the mean term is the "1st term" (i.e. after the current term). I know how to solve "simple" recurrence relations. For instance, say you have:  $$c_0 = 20$$ $$c_1 = 30$$ $$c_n = 3 c_{n-1} - 2 c_{n-2}$$  We can write the characteristic equation as:  $$3x^{n-1} - 2x^{n-2} = x^n$$  Solving this with $n=2$, we get $x = 1$ or $x = 2$. This lets us write the relation $c_n = \alpha_1 1^n + \alpha_2 2^n$, and we can solve for $\alpha_1$ and $\alpha_2$ with the initial states $c_0$ and $c_1$.  However, this depends on the fact that $3x^{n-1} - 2x^{n-2} = x^n$ has two roots.  Now, I'm stuck on another problem where the characteristic equation has fewer roots than terms.  Say I have this recurrence relation instead:  $$a_0 = 0$$ $$a_1 = 2$$ $$a_2 = −1$$ $$a_n = 9a_{n-1} - 15a_{n-2} - 25a_{n-3}$$  The characteristic equation would be:  $$9x^{n-1} - 15x^{n-2} - 25x^{n-3} = x^n$$  However, solving with $n=3$, we only get two roots: $x=-1$ or $x=5$. There are not enough roots to write a relation in the form of $a_n = \alpha_1 r_1^n + \alpha_2r_2^n + \alpha_3r_3^n$. How do I proceed? How to change Blender's default startup settings? How can I change Blender from the default cube/lamp/camera default setup? Also, is it possible to change other features, such as the default view direction (Top, Side, etc) or the Editor type (3D View, Node Editor, Graph Editor, etc)? How to setup a restricted SFTP server on Ubuntu? Getting server name during servlet initialization How can I give a user one password for normal usage when logging in, and another password for system administration and sudo access? I want one user to have two passwords. Can I use a custom Splash screen? The default Splash screens are cool, but I'd like to add some of my own work to remind me of how awesome I am (no, not really).    Is it possible to customize/change the splash screen? Link broken on desktop but work on nautilus file manger I created a shortcut to a text file called &quot;imp&quot; on right-clicking it from the default file manager everything works well. But on clicking it on the desktop I get this  Any Idea why? I want to display the pages 'famous' and 'mschief'.  Using this code, WP_query() doesn't return any result:  $args = array( 'post_type' =&gt; 'page', 'pagename' =&gt; 'famous,mschief', 'posts_per_page' =&gt; 3 );   This code returns one of the 2 pages:  $args = array( 'post_type' =&gt; 'page', 'pagename=famous,mschief', );   And this code returns the 2 pages correctly:  $args = array( 'post_type' =&gt; 'page', 'pagename=famous,mschief', 'posts_per_page' =&gt; 3 );   Why does the first piece of code (the only correct one to me) doesn't work? And why the two others produce these results? How to execute two mysql queries as one in PHP/MYSQL? SQL To search the entire MS SQL 2000 database for a value How to wipe and install ubuntu? Now that I have Google Chrome all set up, I got myself a new laptop. Does anyone have any idea how to transfer my settings, cookies, favorites, etc?   I checked documents and settings\me\application data, and found nothing. This started to happen immediately after I had rebooted the first time after doing a system upgrade. It first starts with a dialogue that says "System program problem detected". Then when I try to hit 'report problem' not much happens. I am led through a dialogue that always ends up the problem cannot be solved.  I am aware this is not a lot of information, however I'm not sure which information I need to publish and how should I obtain it to debug this problem.  Here's a screenshot! Does a logistic regression maximizing likelihood necessarily also maximize AUC over linear models? How to secure database passwords in PHP? When a PHP application makes a database connection it of course generally needs to pass a login and password. If I'm using a single, minimum-permission login for my application, then the PHP needs to know that login and password somewhere. What is the best way to secure that password? It seems like just writing it in the PHP code isn't a good idea. I am trying to find all the Sylow 2 subgroups of S4 using Sylow’s theorems. Now, I know that a Sylow 2 subgroup of S4 has size 8, and that there are either 1 or 3 of them (as the number of of Sylow 2-subgroups has form 1+2k and divides 3, the index).  Now my lecturer states “stabilisers of the 3 different bisections of {1,2,3,4} yield 3 distinct Sylow 2 subgroups”  Now my questions are:  1) What does it mean by stabilisers of 3 different bijections of {1,2,3,4}? Is that to say the elements of S4 which send {1,2} and {3,4} to {1,2},{3,4} i.e. by (12)(34)?  2) How does he know that there are 8 elements of this set? Perhaps this will become clearer after the 1st question is answered.  Many thanks, group theory is hard. Running a script during booting/startup; init.d vs cron @reboot I am currently trying to understand the difference between init.d and cron @reboot for running a script at startup/booting of the system.   The use of @reboot (this method was mentioned in  by hs.chandra) is some what simpler, by simply going into crontab -e and creating a @reboot /some_directory/to_your/script/your_script.txt and then your_script.txt shall be executed every time the system is rebooted. An in depth explanation of @reboot is   Alternatively by embedding /etc/init.d/your_script.txt into the second line of your script ie:  #!/bin/bash # /etc/init.d/your_script.txt  You can run chmod +x /etc/init.d/your_script.txt and that should also result for your_script.txt to run every time the system is booted.  Q1: What are the key differences between the two? Q2: Which is more robust? Q3: Is there a better one out of the two? Q4: Is this the correct way of embedding a script to run during booting?  I will be incorporating a bash .sh file to run during startup.