How do i find sum   $$\sum_{r=1}^{n}\sin(3^{r-1}\theta)\sec(3^r\theta) $$  Needs hint to begin  thanks
The question asks to prove that -  $$\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x}  = \frac 12 (\tan 27x - \tan x) $$  I tried combining the first two or the last two fractions on the L.H.S to allow me to use the double angle formula and get $\sin 6x$ or $\sin 18x$ but that did not help at all.   I'm pretty sure that if I express everything in terms of $x$, the answer will ultimately appear but I'm also certain that there must be another simpler way.
Invalid signature for Kali Linux repositories : "The following signatures were invalid: EXPKEYSIG ED444FF07D8D0BF6 Kali Linux Repository" Display header and footer in all PDF pages using VF My vf page     &lt;apex:page renderAs="pdf" Controller="testpdf1"&gt;     &lt;head&gt;         &lt;style type="text/css" media="print"&gt;             @page              {                 size: 8.5in 11in;/* width height */             }              @page              {                 @top-center                  {                     content: element(header);                 }                  @bottom-left                  {                     content: element(footer);                 }             }              div.header              {                 padding: 10px;                 position: running(header);             }              div.footer              {                 display: block;                 padding: 5px;                 position: running(footer);             }              .pagenumber:before              {                 content: counter(page);             }              .pagecount:before              {                 content: counter(pages);             }         &lt;/style&gt;     &lt;/head&gt;      &lt;div class="header"&gt;             &lt;div align="right"&gt;Header&lt;/div&gt;     &lt;/div&gt;      &lt;div class="footer"&gt;         &lt;div&gt;Page &lt;span class="pagenumber"/&gt; of &lt;span class="pagecount"/&gt;&lt;/div&gt;     &lt;/div&gt;      &lt;div class="content"&gt;         &lt;p&gt;             &lt;apex:repeat value="{!opp}" var="item"&gt;                 &lt;tr&gt;                     &lt;td class="tableContent"&gt;{!item.Name}&lt;/td&gt;                 &lt;/tr&gt;             &lt;/apex:repeat&gt;         &lt;/p&gt;     &lt;/div&gt; &lt;/apex:page&gt;   Controller  public class testpdf1 {     public list&lt;Account&gt; opp{get;set;}      public testpdf1()     {         opp=[SELECT Name  from Account limit 50];     } }   I am getting all records in pdf format but able to get header in first page only and i am not getting header from second pages how can i display header to all pages. I have been wanting to understand how to find the sum of this series.   $$1^p + 2^p + 3^p +{\dots} + n^p$$  I am familiar with Gauss' diagonalised adding trick for the sum of the first $n$ natural numbers.  I can prove the formulas for  $$\begin{align}   \sum_{1}^{n} k^2 &amp;= \frac{n(n+1)(2n+1)}{6}\\  \sum_{1}^{n} k^3 &amp;= \frac{n^2(n+1)^2}{4}\\ \sum_{1}^{n} k^4 &amp;= \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30} \\ \end{align}$$  With mathematical induction. But, beyond that even proofs with mathematical induction are difficult.  I'm interested in learning the theory and the proof behind Faulhaber's formula. What is the knowledge required to understand this proof ? How do I switch back to python2 after Anaconda set python3 as the default? Recently, I installed  on my 15.04 and ended up with this!  :~$ python --version Python 3.5.1 :: Anaconda 2.5.0 (64-bit)  :~$ python2 --version Python 2.7.9  :~$ python3 --version Python 3.5.1 :: Anaconda 2.5.0 (64-bit)   During the last moments of installation, I did enter something as yes in hurry, and I suppose it had to do something with this. (Ok, my fault, I should have handled that carefully, but I need help now, not criticism).    AFAIK this is definitely going to break other programs. What do I do now?    What I think could work.  Can this be done using aliases?  alias python=python2   But I ain't sure. Partition of a set and Hall's theorem ^x^y unix trick for all instances in last command? One of my favorite Unix tricks is ^x^y, which will take the last command and replace the first instance of "x" with "y". However, I'm wondering if a similar trick works to replace all instances of "x" with "y" in the last command? What lifetime can be expected of the typical hard disk? Or are there big differences between different types? And does it make a difference if it is used heavily instead of never being connected to system (for example serving as a backup medium)? Automatically use specific language in Notepad++ for files with no extension How long can I wait before choosing to counter a spell? Adjective, if adjective: eg "pointed, if fatherly" They were pointed, if fatherly, remarks that echoed the themes he has stressed in his papacy but ones that resonated all the more in a newly renovated cathedral surrounded by the luxurious shops of Fifth Avenue. —   I'm not sure the meaning of the phrase &quot;if fatherly&quot; in the above sentence. I'm not familiar with the syntax using the structure 'adjective, if adjective'. How would I go about creating a 3D MMORPG Mouse Look? How would I go about creating a 3D MMORPG Mouse Look? I have created a 3D Space Rpg that is almost finished with success so far. However, I didn't have a clue, how I would attempt to make a Mouse Look method where if a user right clickes and then holds the right mouse button and moves it around the camera moves around with the mouse. The camera can not move away from your character/ship, it should act as if the character/ship is it's origin and that it rotates all around the origin.  I don't mind coding this with python either. Advice is welcome! Hi I would like to know whether the trace of the inverse of a symmetric positive definite matrix $\mathrm{trace}(S^{-1})$ is convex.  Actually I know that the trace of a symmetric positive definite matrix $S\in M_{m,m}$ is convex since we can find $B\in M_{n,m}$ such that $S=B^T\times B$ then we can write the trace as the sum of scalar quadratic forms, i.e. $\mathrm{trace}(S)=\mathrm{trace}(B^T\times B)=\sum_{j=1}^mb_j^T\times b_j$ where $b_j$ is the $j^{th}$ column of $B$.  for instance if we have  $trace([\begin{array}{cc} 1 &amp; 2  \\ 3 &amp; 4  \\ \end{array}] \times [\begin{array}{cc} 1 &amp; 3  \\ 2 &amp; 4  \\ \end{array}])= [\begin{array}{cc} 1 &amp; 2  \\ \end{array}]\times [\begin{array}{c} 1   \\ 2   \\ \end{array}]+ [\begin{array}{cc} 3 &amp; 4  \\ \end{array}]\times [\begin{array}{c} 3   \\ 4   \\ \end{array}]=30$  And so I wonder if $\mathrm{trace}(S^{-1})$ is convex too.. Is it possible to reset all folder permissions? I was wondering if it was possible to reset all the folders' rights. I was having trouble when trying to install LAMP on my Ubuntu 11.04 installation, and I modified many permissions like /var /etc and I want to cancel everything I did and have permissions set to how they would be after a new install. Fastest converging method for calculating e Schrödinger Equation for Two Electrons? I'm having trouble finding a simple answer to this question (maybe because there isn't one), but I'm just confused about how the Schrödinger Equation would look for two electrons. I understand that it would exist in 6 dimensional configuration space, but how does the potential look? It's confusing that the potential would be different for each electron depending on where the other one is. Consider the following Gaussian Integral $$I = \int_{-\infty}^{\infty} e^{-x^2} \ dx$$  The usual trick to calculate this is to consider $$I^2 = \left(\int_{-\infty}^{\infty} e^{-x^2} \ dx \right) \left(\int_{-\infty}^{\infty} e^{-y^{2}} \ dy \right)$$  and convert to polar coordinates. We get $\sqrt{\pi}$ as the answer.   Is it possible to get the same answer by considering $I^{3}, I^{4}, \dots, I^{n}$? I have a question related to multiple regression and interaction, inspired by this CV thread:    When checking for a moderation effect I do center my independent variables and multiply the centered variables in order to calculate my interaction term. Then I run my regression analysis and check for main and interaction effects, which may show the moderation.  If I redo the analysis without centering, apparently the coefficient of determination ($R^2$) does not change but the regression coefficients ($\beta$s) do. That seems clear and logical.  What I do not understand: The p-values of the main effects change substantially with centering, although the interaction does not (which is right). So my interpretation of main effects could change dramatically - just determined by centering or not. (It's is still the same data, in both analyses!)  Can somebody clarify? - Because that would mean that the option to center my variables would be mandatory and everybody should do it in order to get the same results with the same data.    Thanks a lot for distributing to that problem and your comprehensive explanations. Be assured that your help is very much appreciated!  For me, the biggest advantage of centering is to avoid multicollinearity. It is still quite confusing to establish a rule, whether to center or not. My impression is, that most of the resources suggest to center, although there are some "risks" when doing it. Again I want to put out the fact, that 2 researchers dealing with the same material and data might conclude different results, because one does centering and the other does not. I just read some part of a book by Bortz (he was a Professor and kind of a Statistics Star in Germany and Europe), and he does not even mention that technique; just points out to be careful in interpreting main effects of variables when they are involved in interactions.  After all, when you conduct a regression with one IV, one moderator (or second IV) and a DV, would you recommend to center or not? I was trying to experiment with docker, Ubuntu 14.04, 64-bit. I managed to run an image, and I gave it the path on one of my removable drives, /media/Drive1.  Now, whenever I reboot, I see there is a /media/Drive1 with only a docker directory inside it, and the actual drive which should be Drive1 ends up mounted on /media/Drive11, and this happens every reboot - even when I unmount these drives, and do a sudo rm -rfv /media/Drive1 before rebooting!  So something probably creates /media/Drive1 before it is fully mounted, and I guess that something is docker, because when I boot, I have:  $ ps axf | grep docker  2085 ?        Ssl    0:00 /usr/bin/docker -d   So, how can I disable the docker autostart, so it doesn't interfere with the mounting of external USB drives at boot? Add UI elements to UIScrollView? A sphere of radius R is rolling without slipping with a velocity v and collides inelastically with a step of height h &lt; R. What is the minimum velocity for which the sphere will be over the step?    Relevant equations  Total kinetic energy (maybe): $\frac{1}{2}I\omega^{2}+\frac{1}{2}mv^2$  Gravitational potential energy: $mgh$   Moment of inertia about a point a distance d from the CM from de center of mass: $I=I_{cm}+md^{2}$ Rolling without slliping  $v=\omega R$  A discussion of the problem This problem is confusing to me: I know the collision is inelastic so there is no conservation of kinetic energy, there is torque when the sphere pivots the step so the angular momentum is not conserved either, the linear momentum doesn't give a lot information and even though I tried to see what condition needs the torque about the pivot point (to give a net torque in the direction of rotation and thus passing the step), I'm not able to see which torque can overcome the one from gravity which doesn't let the sphere slide up:    The sphere has an angular velocity, but I'm missing something conceptual because I can't figure out how to formally state that the angular momentum changed the right amount so that the sphere made it past the step.  The attempt at a solution  My (almost certainly wrong) attempt was to say that the sphere lost kinetic energy because of the change on potential energy:  $\frac{1}{2}I\omega^{2}+\frac{1}{2}mv^2 +mgR = mg(R+h)\\ v=\sqrt{\frac{10}{7}gh} $  I hope you can help me :)