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Properties of $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$ where $\{a_n\}$ is unbounded, strictly increasing sequence of positive reals Let $\{a_n\}$ be an unbounded, strictly increasing sequence of positive real numbers and let $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$. Which of the following statements is/are correct? (CSIR NET December 2014)         For all $n\geq m, \sum\limits^{n}_{k=m}x_k&gt;1-\bf{\frac{a_m}{a_n}}$   There exist $n\geq m, \sum\limits^{n}_{k=m}x_k&gt;\frac{1}{2}$      My attempt:  Analysing question in light of $a_k=k\implies x_k=\frac{k+1-k}{k+1}=\frac1{k+1}$, I feel both are correct, I doubt if  I can generalise the result in light of this one example.   Any way to prove that this true for general case?    Note: Please do not confuse  with my question. The first two options are different, I've marked the difference in bold. Output for Print Is there a way to output for Print. I want to save in a higher DPI than the default 72dpi say 300dpi that's a good quality for printed pieces. I tried downloading an add-on but don't know if I am installing it correctly it isn't showing up but the bar in python is green. I'm using the 2.69 version of Blender. If no such feature exists what size have any of you found best for print quality. Thank you Possible A* Optimizations As I have heard people did not trust Euler when he first discovered the formula (solution of the ) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs.   I believe many of you know some nice proofs of this, can you please share it with us? What's wrong with this experiment showing that either FTL communication is possible or complementarity doesn't hold? The assumptions are:    Alice and Bob have perfectly synchronized clocks Alice and Bob have successfully exchanged a pair of entangled photons   The idea is simply to have Alice and Bob perform the Quantum Eraser Experiment (doesn't need to be the delayed choice).   Alice and Bob agree on a specific time when Bob's photon will be between the "path marker" (which is usually just after the slits) and the detector.  If Alice acts collpasing the wave-function on her photon, the interference pattern will disappear. If not it won't.  Alice and Bob can be spatially separated...  What am I misunderstanding?   The only meaningful difference from this spatially separated quantum eraser experiment to one done on tabletop is that you won't be able to use a coincidence detector, but that is not impeditive to identifying the interference pattern, just will make errors more probable. Which we should be able to deal with a appropriate protocol...  There is a experimental paper with a small amount of citations pointing out to the breaking of complementarity in a very similar setup: When we lie down, the "back part of the body" is in contact with the bed. It includes the heels, the calves, the backside, the back, and the back of the neck, and the back of the head etc.  How do we refer to the entire "back part" of the body? Some Data Failed to Reconstruct, Motion Tracking I added tracks and stuff, but then when I tried to solve motion track, there was a error that said:     Some Data Failed to Reconstruct   and it also said to look into the console for details, which I did. This is what the console said:    By my understanding, my video is 372 frames long, and Blender is cutting the footage at 318 for some reason. Here is the image of the scene:    Why? Here are links to  and also . Also the next morning I did a camera solve, but then most of the frames went red. Is my model any good, based on the diagnostic metric ($R^2$/ AUC/ accuracy/ RMSE etc.) value? Highlighting the selected tab using jQueryUI Tabs I have a header page with links to all the visualforce page as tabs.I need to highlight the tab on select of it.      &lt;apex:page title="{!$Label.site.site_login}" showHeader="false" sidebar="false"       standardStylesheets="false"&gt; &lt;apex:includeScript value="{!URLFOR($Resource.jQuery, '/js/jquery-1.4.2.min.js')}"  /&gt;   &lt;apex:includeScript value="{!URLFOR($Resource.jQuery, '/js/jquery-ui-1.8.6.custom.min.js')}"  /&gt; &lt;apex:stylesheet value="{!URLFOR($Resource.jQuery, '/css/ui-lightness/jquery-ui-1.8.6.custom.css')}"  /&gt;          &lt;html lang="en"&gt;  &lt;head&gt;   &lt;script src="http://ajax.googleapis.com/ajax/libs/jqueryui/1.8/jquery-ui.min.js"&gt;  $(function()  {   $( "#tabs" ).tabs();   });  &lt;/script&gt;   &lt;style&gt;   #tabs .ui-tabs-active    { background: red; }  &lt;/style&gt;     &lt;/head&gt;      &lt;body&gt;         &lt;nav class="navbar navbar-static-top navbar-inverse" role="navigation"&gt;             &lt;div class="container"&gt;                 &lt;div class="navbar-header"&gt;                     &lt;button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-ex1-collapse"&gt;                         &lt;span class="icon-bar"&gt;&lt;/span&gt;&lt;span class="icon-bar"&gt;&lt;/span&gt;                         &lt;span class="icon-bar"&gt;&lt;/span&gt;                      &lt;/button&gt;                     &lt;a class="navbar-brand" style="line-height: 0px; padding: 10px 15px;" href="index.php.html"&gt;                         &lt;img height="30px" src="{!URLFOR($Resource.Resources, 'img/logo.png ')}" /&gt;                     &lt;/a&gt;                 &lt;/div&gt;                 &lt;div class="collapse navbar-collapse navbar-ex1-collapse" id="tabs"&gt;                     &lt;ul class="nav navbar-nav"&gt;                         &lt;li id="Home"&gt;                             &lt;apex:outputLink value="{!$Page.Home}"&gt;Home&lt;/apex:outputLink&gt;                         &lt;/li&gt;                         &lt;li id="chatter"&gt;                             &lt;apex:outputLink value="{!$Page.Chatter}"&gt;Chatter&lt;/apex:outputLink&gt;                         &lt;/li&gt;                         &lt;li id="page3"&gt;                             &lt;apex:outputLink value="{!$Page.page3}"&gt;Room&lt;/apex:outputLink&gt;                         &lt;/li&gt;                         &lt;li id="page4"&gt;                             &lt;apex:outputLink value="{!$Page.page4}"&gt;Page4&lt;/apex:outputLink&gt;                         &lt;/li&gt;                     &lt;/ul&gt;                 &lt;/div&gt;             &lt;/div&gt;         &lt;/nav&gt;     &lt;/body&gt;      &lt;/html&gt; &lt;/apex:page&gt; What to do when someone other than the author changes a question to fix a substantive problem with the code See . Someone other than the OP changed the code to address a substantial problem with the it that was addressed as part of my answer. The problem was admittedly peripheral to the core question, but it was a primary differentiator between my answer and another answer submitted at the same time.  Any suggestions on what to do in this situation? Leave it be? Edit it back? Something else?  Update: I just found , from the "Related Links" section on the right, which is pretty relevant. I have to keep remembering to check there rather than just searching for answers. Promote tag wiki creation for new tags Positioning of decorations on parallel lines I'm displayed three lines on a pgfplot. I want to show a marker for the slope of each line and the difference in height between lines.     I'm showing the slopes by adding triangles under the lines  (at a manual position) and placing an annotation inside the triangle (at a manual position). I'm showing the difference between the height of the lines by adding a brace manually as well.   How can I place the braces on the plot and anchor them at a certain point on each line? The triangles can be added at the end of each line declaration, but doing it like this is ok. The braces are put in place manually, which is not sustainable as I create more plots.   I want to avoid using xshift, yshiftand manual adjustment to put the braces,traingles and annotations in the right place. How can I do this?      \documentclass{article} \usepackage{pgfplots}  \pgfplotsset{compat=1.9}  \begin{document}   \begin{tikzpicture}[scale=0.45,trim axis left, trim axis right]     \begin{axis}[ylabel={$\hat{S(4)}$},xlabel={$S{^*}_{breakeven}$},ylabel style={rotate=-90},domain=1:25, legend style={legend pos=north west,font=\scriptsize}, legend cell align=left]  \addplot+[samples=200,mark=none]                                    {6.26 + 0.28*x} coordinate[pos=0.5](x) coordinate[pos=0.9] (y) (x)-| (y) node [midway,xshift=-0.6cm,yshift=0.3cm] {\footnotesize $Slope X$};;  \addplot+[samples=200,mark=none, dotted, thick]             {2.66 + 0.28*x}  coordinate[pos=0.3](a) coordinate[pos=0.6] (b) (a)-| (b) node [midway,xshift=-0.5cm,yshift=0.2cm] {\footnotesize $Slope 1$};  \draw [decorate,decoration={brace,amplitude=5pt,raise=4pt},xshift=4pt]  (225,107) -- (225,73)  node [midway,xshift=0.6cm] {\footnotesize $\beta_1$};  \end{axis} \end{tikzpicture} \end{document} Why do I have the "required reviews completed" status on my submission in Elsevier? I had submitted my paper in an Elsevier journal one year back. Now, status update on the submission tracking says "Required reviews completed." However, for the last two weeks still there is no change in the status.  Should I ask the editor what is going on with my submission? How do I tell if matrices are similar? I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not.  I solved this by using a matrix called $S$: $$\left(\begin{array}{cc} a&amp; b\\ c&amp; d \end{array}\right)$$ and its inverse in terms of $a$, $b$, $c$, and $d$, then showing that there was no solution to $A = SBS^{-1}$. That worked fine, but what will I do if I have $3\times 3$ or $9\times 9$ matrices? I can't possibly make system that complex and solve it.  How can I know if any two matrices represent the "same" linear transformation with different bases?  That is, how can I find $S$ that change of basis matrix?  I tried making $A$ and $B$ into linear transformations... but without the bases for the linear transformations I had no way of comparing them.  (I have read that similar matrices will have the same eigenvalues... and the same "trace" --but my class has not studied these yet. Also, it may be the case that some matrices with the same trace and eigenvalues are not similar so this will not solve my problem.)  I have one idea. Maybe if I look at the reduced col. and row echelon forms that will tell me something about the basis for the linear transformation? I'm not really certain how this would work though? Please help. Drag-and-drop functionality no longer working after upgrade to 19.04 After upgrading from 18.04 to 19.04 (with a necessary upgrade to 18.10 between) on a Lenovo Yoga 11e, I can no longer drag and drop anything (ie, files, windows, etc.) with the touchpad.  I can, however, move things around using the touchscreen.  Whenever I attempt to do so with the mouse, though, sometimes it turns into a hand.  Whether or not it turns into the hand or remains a pointer, the icon will not move anymore.  IOW, as soon as I left-click, the pointer freezes in place until I release the touchpad. Calculating $\lim_{n\to\infty}\sqrt[n]{2^n-n^2}$ Highlight whole TreeViewItem line in WPF The external Flash is not showing up in the configuration Menu when using Pixel TF-321 hot shoe. Is this the wrong adapter for that camera, as there is no X connection? I want to trigger a studio flash via sync connection. Can you recommend me a working hot shoe? Thanks! How do I prevent Eclipse from hanging on startup? I am using Eclipse 3.3 ("Europa"). Periodically, Eclipse takes an inordinately long time (perhaps forever) to start up. The only thing I can see in the Eclipse log is:       !ENTRY org.eclipse.core.resources 2 10035 2008-10-16 09:47:34.801     !MESSAGE The workspace exited with unsaved changes in the previous session; refreshing workspace to recover changes.   Googling reveals  that I remove the folder:  workspace\.metadata\.plugins\org.eclipse.core.resources\.root\.indexes   This does not appear to have helped.  Short of starting with a new workspace (something which I am not keen to do, as it takes me hours to set up all my projects again properly), is there a way to make Eclipse start up properly? Determinant of a square matrix in a field \begin{array}{rrrrr|r}     b &amp; a &amp; a &amp; \cdot \cdot \cdot &amp; a \\     a &amp; b &amp; a &amp; \cdot \cdot \cdot &amp; a \\     a &amp; a &amp; b &amp; \cdot \cdot \cdot &amp; a \\     \cdot &amp; \cdot &amp; \cdot &amp; \space &amp; \cdot\\ \cdot &amp; \cdot &amp; \cdot &amp; \space &amp; \cdot\\ a &amp; a &amp; a &amp; \cdot \cdot  \cdot &amp; b   \end{array}  I have the above matrix $A\in M_{n\times n}(F)$ where $F$ is a field and $n\geq1$, $a,b\in F$.  I'm trying to find out how to use row operations to make it into an upper triangular matrix in order to figure out the determinant. But I'm not sure how I would approach it.