Let $n$ be a natural number and let $\langle n\rangle$ denote the closest integer to $\sqrt n$. Evaluate $$\sum_{n=1}^  \infty \frac{2^{\langle n\rangle}+2^{-\langle n\rangle}}{2^n}$$ I started putting the values of $n$ to see a pattern. It looks like a Geometric Progression but I am unable to sum it. Also, if a better approach exists, please share. Thanks.
How can I obtain the limit of the series     $$\sum_{n=1}^{\infty} \frac{2^{[\sqrt{n}]}+2^{-[\sqrt{n}]}}{2^n}$$   Where $[\ \ ]$ is .
Is there an unsubdivide function in Blender? I am wondering if there is a modifier or tool that will essentially undo a subsurf or subdivide command. Distribution of orbital velocities in a disk galaxy for N-body simulation? I am working on an iGoogle-like application. Content from other applications (on other domains) is shown using iframes.   How do I resize the iframes to fit the height of the iframes' content?  I've tried to decipher the javascript Google uses but it's obfuscated, and searching the web has been fruitless so far.  Update: Please note that content is loaded from other domains, so the  applies. Send Gmail calendar invite from alternate email After plain installation of Ubuntu 16.04 LTS gnome-software doesn't work. It has opened once with GUI, but installation of Unity Tweak Tool didn't finish.  When starting again, GUI doesn't come up. When opening from terminal, a warning appears, but the application GUI doesn't open.  $ gnome-software (gnome-software:7052): Gs-WARNING: failed to open plugin /usr/lib/gs-plugins-9/libgs_plugin_xdg_app_reviews.so: /usr/lib/gs-plugins-9/libgs_plugin_xdg_app_reviews.so: cannot open shared object file: No such file or directory   Does anybody have same experience, and eventually, a solution?    According to the recommendation:  $ dpkg-query -S /usr/lib/gs-plugins-9/libgs_plugin_xdg_app_reviews.so gnome-software: /usr/lib/gs-plugins-9/libgs_plugin_xdg_app_reviews.so   Proposed reinstall of gnome-software did not help.    As workaround, use Synaptic Package manager and command line. How do I read a text file with variable hexadecimal value columns in matlab? I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded into the Euclidean space of the same dimension. So I wonder if all manifolds can be embedded into a same dimensional Euclidean space when we cut out a measure zero set? Can anyone prove it or disprove it by giving me some counterexamples? This question was prompted by . Notoriously, if you calculate the Schwarzschild coordinate time for anything, matter or light, to reach the event horizon the result is infinite. This implies that the universe ages by an infinite time before someone falling into the black hole reaches the event horizon, so could that person see the universe age by an infinite time?  To be more precise, suppose the observer starts falling from rest at time $t = 0$ and some initial distance $r &gt; r_s$. If we wait for some time $T$ then shine a light ray at the falling observer. Will the light ray always reach the falling observer before they cross the event horizon? If not, what is the formula for the longest time $T$ that we can wait and still be sure the ray will catch the observer? If $T$ is not bounded it implies that observer could indeed see the end of the universe.  I can think of a qualitative argument for an upper limit on $T$, but I'm not sure how sound my argument is. The proper time for the observer to fall to the event horizon is finite - call this $\tau$. The proper time for the light ray to reach the horizon is zero, therefore the light ray will reach the observer before they cross the event horizon only if $T &lt; \tau$. Hence $T$ is bounded and the observer won't see the end of the universe.  I think a more rigorous approach would be to determine the equations of motion (in the Schwarzschild coordinates) for the falling observer and the light ray, and then find the condition for the light to reach the falling observer at some distance $\epsilon$ from the event horizon. Then take the limit as $\epsilon \rightarrow 0$. In principle this seems straightforward, but in practice the algebra rapidly defeated me. Even for a light ray the radial distance:time equation isn't closed form (Wolfram claims it needs the $W$ function) and for the falling observer the calculation is even harder. How to add a program as a startup application from a terminal? I am using Ubuntu 12.04 and I know the GUI way to maintain the startup program list but now I want to find out the way to add/remove startup applications using a command line interface (CLI). How do I perform a Perl substitution on a string while keeping the original? How to calculate Area Under the Curve (AUC), or the c-statistic, by hand I am interested in calculating area under the curve (AUC), or the c-statistic, by hand for a binary logistic regression model.  For example, in the validation dataset, I have the true value for the dependent variable, retention (1 = retained; 0 = not retained), as well as a predicted retention status for each observation generated by my regression analysis using a model that was built using the training set (this will range from 0 to 1).  My initial thoughts were to identify the "correct" number of model classifications and simply divide the number of "correct" observations by the number of total observations to calculate the c-statistic. By "correct", if the true retention status of an observation = 1 and the predicted retention status is > 0.5 then that is a "correct" classification. Additionally, if the true retention status of an observation = 0 and the predicted retention status is &lt; 0.5 then that is also a "correct" classification. I assume a "tie" would occur when the predicted value = 0.5, but that phenomenon does not occur in my validation dataset. On the other hand, "incorrect" classifications would be if the true retention status of an observation = 1 and the predicted retention status is &lt; 0.5 or if the true retention status for an outcome = 0 and the predicted retention status is > 0.5. I am aware of TP, FP, FN, TN, but not aware of how to calculate the c-statistic given this information. Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field Is there a way to force the Nikon D40's rear LCD panel to stay off? Is there an off/disable control anywhere? I realize that the INFO button toggles it off, but it comes right back on as soon as you tap the shutter, and that it a problem in low light situations where the LCD keeps blasting me in the eye. guess the color of the next card There are 26 red cards and 26 black cards on the table which are randomly shuffled and are facing down onto the table. The host turns up the cards one at a time. You can stop the game any time (even at the beginning of the game). Once you stops the game, the next card is turned up: if it is red, you get $1; otherwise you pay the host one dollar. What is the payoff of this game? Which parts are from Eminems real life? In "" we saw how Bunny Rabbit got into the rapping gang and won the heart of millions of rap lovers. Now in Eminem's original life too similar thing happened. We know how Dr. Dre helped him getting into an industry which is dominated by mainly black people in USA. In fact,  called the film semi-autobiographical. So I want to know which parts are direct picturisations of his real life? Is re-projection needed from SRID 4326 (WGS 84) to SRID 4269 (NAD 83)? I have US census data against state boundaries stored as  SRID 4269, in an MSSQL DB.  The input data used for calculations against that dataset is stored as SRID 4326.  As far as I can infer from the , SRID 4269 is just a subset of SRID 4326 and no coordinate re-projection is necessary.  Do I actually need to re-project from SRID 4326 to SRID 4269, provided that the input data is always within the bounds of the SRID 4269? ls: cannot access .gvfs: Permission denied Whenever I run an application in terminal as root (such as sudo gedit  /etc/default/varnish) , subsequently when I open another terminal I get "ls: cannot access .gvfs: Permission denied" error at top line of terminal.   I found a solution on the net     umount /path/to/.gvfs   rm -rf .gvfs   but it only fixes problem temporarily.  It appears that I have two mounted instances of gvfs in my system    $ sudo mount |grep gvfs     gvfsd-fuse on /run/user/1000/gvfs type fuse.gvfsd-fuse (rw,nosuid,nodev,relatime,user_id=1000,group_id=33)     gvfsd-fuse on /home/****/.gvfs type fuse.gvfsd-fuse (rw,nosuid,nodev,relatime,user_id=0,group_id=0)   I don't know if it's related to, a few months ago I had to change permission in my home folder like     sudo chown -R $USER:www-data   Would you please help me fix it?  EDIT: After I unmount /run/user/1000/gvfs I don't get that error. I usually disconnect my cellphone from the charger some time after it's fully charged because I'm afraid the battery could lose capacity over time if I don't. However, I believe modern electronics should be able to handle this kind of scenario and automatically stop charging the battery once it's full. Does this apply to modern Android phones? Or do I need to continue disconnecting it all the time? What is the width of the standard paragraph indent in LaTeX's article class? If no changes has been made to the settings of indentation of an article class LaTeX document, what is the width of the paragraph indent?  texdoc article in the terminal opens the Standard Document Classes for LaTeX version 2e manual. In section 6.2 Paragraphing I find the following line:  ⟨10pt⟩ \setlength\parindent{15\p@}   The  means 1pt ... so I guess this would mean that the standard indent is 15 pt at a 10 pt text size? Is this correct? What does the explicit keyword mean? What does the explicit keyword mean in C++?