Calculate the value of $\frac{1}{2}+\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+....+\frac{F_n}{2^n}+....$
How to prove the Fibonacci sum $\sum \limits_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$
round() doesn't seem to be rounding properly Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$? What does this bag with a checkmark on it next to Roblox usernames mean? VirtualBox 5.0.40 crashes the whole host since update to kernel 4.13.0-26-generic I received the updates through normal Software Updater, 10-Jan-2018. After this VirtualBox would not start any VM, complaining about dkms and asking to make modprobe vboxdrv. Following the advice I reinstalled virtualbox-dkms with:  sudo apt-get --reinstall install virtualbox-dkms   After this and a reboot I tried to start a VM which freezed the whole machine and a hard reboot was necessary.  Is there a way to make VirtualBox work with the latest kernels?  EDIT:  As suggested by N0rbert, the command:  apt-cache policy `dpkg -S VirtualBox | head -n1 | sed -e 's/://g' | awk '{print $1}'`  gives:   virtualbox-qt:   Installed: 5.0.40-dfsg-0ubuntu1.16.04.2   Candidate: 5.0.40-dfsg-0ubuntu1.16.04.2   Version table:  *** 5.0.40-dfsg-0ubuntu1.16.04.2 500         500 http://fr.archive.ubuntu.com/ubuntu xenial-updates/multiverse amd64 Packages         100 /var/lib/dpkg/status      5.0.18-dfsg-2build1 500         500 http://fr.archive.ubuntu.com/ubuntu xenial/multiverse amd64 Packages   EDIT (on 29-Mar-2018): Issue is resolved for me, with the Ubuntu Ubdate from today. I have seen that it had some virtualbox-updates, therefore I tried to boot with the most recent kernel again. Then I had to run  sudo apt-get --reinstall install virtualbox-dkms  And after this my virtual machines in VirtualBox started working without crashing the whole computer. Finally! Display solution/file path in the Visual Studio IDE I frequently work with multiple instances of Visual Studio, often working on different branches of the same solution.   used to display the full path of the current source file in its title bar, but Visual&nbsp;Studio 2005 doesn't appear to do this. This makes it slightly more awkward than it should be to work out which branch of the solution I'm currently looking at (the quickest way I know of is to hover over a tab so you get the source file's path as a tooltip).  Is there a way to get the full solution or file path into the title bar, or at least somewhere that's always visible, so I can quickly tell which branch is loaded into each instance? I want to find out the number of possible combinations of $x$ numbers that sum to $y$. For example, I want to calculate all combination of 5 numbers, which their sum equals to 10.  An asymptotic approixmation is also useful. This question seems to be very close to number partitioning, with the difference that a number can be 0. See:    All possible partitions for sum 10 and 3 positions that can be zero, are 63 possiblities: (numbers shown as 3 digits)  019 028 037 046 055 064 073 082 091 109 118 127 136 145 154 163 172 181 190 208 217 226 235 244 253 262 271 280 307 316 325 334 343 352 361 370 406 415 424 433 442 451 460 505 514 523 532 541 550 604 613 622 631 640 703 712 721 730 802 811 820 901 910 Is it possible to remove "Show Desktop" from the Alt+Tab application switcher? What is the exposure triangle? How do the "sides" affect my photographs? What test can I use to compare slopes from two or more regression models? I would like to test the difference in response of two variables to one predictor. Here is a minimal reproducible example.   library(nlme)  ## gls is used in the application; lm would suffice for this example m.set &lt;- gls(Sepal.Length ~ Petal.Width, data = iris,                 subset = Species == "setosa") m.vir &lt;- gls(Sepal.Length ~ Petal.Width, data = iris,                 subset = Species == "virginica") m.ver &lt;- gls(Sepal.Length ~ Petal.Width, data = iris,                 subset = Species == "versicolor")   I can see that the slope coefficients are different:  m.set$coefficients (Intercept) Petal.Width    4.7771775   0.9301727 m.vir$coefficients (Intercept) Petal.Width    5.2694172   0.6508306  m.ver$coefficients (Intercept) Petal.Width     4.044640    1.426365    I have three questions:   How can I test the difference between slopes? How can I test the difference between residual variances? What is a simple, effective way to present these comparisons?    A related question, , suggests re-running the model with a dummy variable to differentiate the slopes, are there options that would allow the use of independent data sets? I don't understand why su - is preferred over su to login as root. When playing as a spellcaster, is it valid the use of the Ready action to prepare Dispel Magic to counter a spell cast by an opponent? In the same vein, when declaring said action, do I have to specify which spell slot I will use when Readying the Dispel Magic spell? If that is the case, does this tactic nullify an opponent's spell? I get the impression that this is basically turning Dispel Magic into a spell with a casting time of &quot;1 reaction&quot;. Is my reading of the rules accurate? A very recent post asking people to go "on strike" was just deleted How do I connect to my Bluetooth headset? I have an ACER C7 Chromebook with Ubuntu 12.04 installed.  Also I have two apps installed Bluetooth Manager and Bluetooth.    I am trying to connect an  to Ubuntu. I have tried to set up the new device and pair, but the headset is never recognized.  I checked hard and soft setting for Bluetooth and they are on.  I am new to Ubuntu 12.04 and Linux and need specific instructions. In this case what is correct?        Keep Tom and I updated.      or         Keep Tom and me updated.      I understand that me is an object pronoun. And therefore I feel the second option is correct.  Especially, because I would say "Keep him updated" and also "Keep me updated". Instead of "Keep he updated" or "Keep I updated". But something about saying "Keep him and me updated" feels wrong.  However I am not sure, because several people have told me that in this case I is the correct option because it is being used in conjunction with a name (Tom) and not a pronoun. Is that correct? How deterministic is floating point inaccuracy? I have two tables in my database:  NEWS table with columns:   id - the news id  user - the user id of the author)   USERS table with columns:   id - the user id   I want to execute this SQL:   SELECT * FROM news JOIN users ON news.user = user.id    When I get the results in PHP I would like to get associative array and get column names by $row['column-name']. How do I get the news ID and the user ID, having the same column name? Using the method of characteristics, find a solution to Burgers' equation    \begin{cases} u_t+\left(\frac{u^2}{2} \right)_x =0 &amp; \text{in }\mathbb{R}\times(0,\infty) \\ \qquad \qquad \, \, u=g &amp; \text{on } \mathbb{R} \times\{t=0\} \end{cases}   with the initial conditions    $$g(x)=\begin{cases} 0 &amp; \text{if }x &lt; 0 \\ 1 &amp; \text{if }0 \le x \le 1 \\ 0 &amp; \text{if }x &gt; 1 \end{cases}$$   First, I realized that the equation $u_t+\left(\frac{u^2}{2} \right)_x =0$ is equivalent to this form: $$u_t+uu_x =0$$  Then should I generally follow the method of solution as outlined in the answer of ?  Note that this is not a duplicate question of that page. Rather I want to know if that page can be used for my problem, even though my ICs are different.  By the way, the solution printed in my book (PDE Evans, 2nd edition, page 142) is      $$u(x,t) =  \begin{cases} 0 &amp; \text{if } x &lt; 0 \\  \frac xt &amp; \text{if } 0 &lt; x &lt; t  \\ 1 &amp; \text{if } t &lt; x &lt; 1 + \frac t2  \\ 0 &amp; \text{if } x &gt; 1 + \frac t2 \tag{$0 \le t \le 2$}  \end{cases}$$ In evaluating integral $\int\sin^{3}x dx $ I am pretty sure we need to use substitution $e^x=t$, but can't go  next step. How do I place a bootable ISO on a USB drive? window-switching keyboard shortcuts interfere with emacs and cannot be changed in the main keyboard system preferences