Series with reciprocal of binomial coefficients
Proving that $\sum_{k=0}^{n}\frac{(-1)^k}{{n\choose k}}=[1+(-1)^n] \frac{n+1}{n+2}.$
Integer reciprocals in arithmetic progression Let $m_1,m_2,\ldots,m_k$ be $k$ positive integers such that their reciprocals are in AP. Show that $k&lt;m_1+2$. Also find such a sequence.     Whatever way I tried, whichever formula I used, I could not eliminate the $m_k$ term, which I reckon would be the ideal scenario to find the solution. I am exhausted of any further ideas, please help. Thank you. Find remainder when $f(x^{12})$ is divided by $f(x)$ How to connect an LED indicator to a circuit? I would like to know how can i add a safety switch with an LED indicator to a circuit , so that if the safety switch is on then the LED lights , and when i want to use the device , all what i have to do is to press the normally-open switch.   I have attached this schematic and im not sure if it will work correctly  ,as i am a very beginner in electronics   Thanks Is the following proof correct, or are certain steps illegal?  Suppose $f:G_1\to G_2$ is a surjective group isomorphism, $N_1\lhd G_1, N_2\lhd G_2$ normal subgroups, with $N_1=f^{-1}(N_2)$. Prove that $(G_1/N_1)\cong (G_2/N_2)$.  Proof:  As $f$ is surjective, we an use the first isomorphism theorem to get $G_1/\ker(f)\cong G_2$, which implies that $$(G_1/\ker f)/(N_1/\ker f)\cong G_2/(N_1/\ker f)$$  Using the first isomorphism theorem we also know that $$N_1/\ker (f)\cong f(N_1)=f(f^{-1}(N_2))=N_2$$  Combining this we can see that $(G_1/\ker f)/(N_1/\ker f)\cong G_2/(N_1/\ker f)\cong G_2/N_2$.  Using the third isomorphism theorem we have that: $$ (G_1/\ker f)/(N_1/\ker f)\cong G_1/N_1$$  And from the last two lines we have $G_1/N_1\cong G_2/N_2$ $\tag*{$\Box$}$ Find no. of non negative integer solutions of $a+2b+3c = 200$ I want to find number of the non negative integer solutions of $a+2b+3c = 200$ (answer can contain $a,b,c = 0$)  Example one of the solutions is $a = 100, b = 50, c = 0$  I used stars and bars trick and here i get is  $\dfrac{\dbinom{200+2}{2}}{2 \times 3}$ but it is incorrect  any type of math is welcomed    what I have learnt from stars and bars technique is to find the solutions of $x+y = n , x,y,n &gt; 0$ are positive integers but I am confused with equation of type $ax+by = n$ where $a , b&gt; 1$ I'm trying to display models that have shading drawn or baked directly on the textures, but I can't figure out how to disable Blender's default scene lighting. If there are no lights in the scene, everything is black, but I want to see the textured objects fully bright. How do I disable the lighting?  NOTE: Just before posting this, I found the "Shadeless" material property, which does more or less what I want. However, is there a way to globally toggle the default lighting? I have a  to which I'd like to add JPEG and PNG images that I generate on the fly.  All the examples I've seen so far in the , specially in the  use ImageIcons.  I'm generating these images as byte arrays, and they are usually larger than the common icon they use in the examples, at 640x480.   Is there any (performance or other) problem in using the ImageIcon class to display an image that size in a JPanel? What's the usual way of doing it? How to add an image to a JPanel without using the ImageIcon class?   Edit: A more careful examination of the tutorials and the API shows that you cannot add an ImageIcon directly to a JPanel. Instead, they achieve the same effect by setting the image as an icon of a JLabel. This just doesn't feel right... What are the major differences between ANSI C and K&R C? How to install CUDA 8.0 on Ubuntu 16.04 with Nvidia GeForce GTX 1080 I am attempting to install CUDA 8.0 within Ubuntu 16.04 for the Nvidia GeForce GTX 1080. I had CUDA 7.5 installed, but it did not appear to be compatible with the GTX 1080. I downloaded the .run installer from Nvidia's developer page, but the installation fails and outputs the following to a nvidia-installer.log     An NVIDIA kernel module 'nvidia-drm' appears to already be loaded in     your kernel.  This may be because it is in use (for example, by an X     server, a CUDA program, or the NVIDIA Persistence Daemon), but this     may also happen if your kernel was configured without support for     module unloading.  Please be sure to exit any programs that may be     using the GPU(s) before attempting to upgrade your driver.  If no     GPU-based programs are running, you know that your kernel supports     module unloading, and you still receive this message, then an error     may have occured that has corrupted an NVIDIA kernel module's usage     count, for which the simplest remedy is to reboot your computer.   I have rebooted the computer, but that did not resolve the problem. How do I unload the 'nvidia-drm' module so that I can install the 8.0 CUDA? Blank screen after installing nvidia restricted driver I've  and rebooted my machine and now I get a black or blank screen.   How can I fix this? Can someone give me a short description and comparison of the different road cycling pedal / shoe systems? I am looking at buying a pedal/shoe system as the clips and straps I am using are hurting my feet again.  It's been a long time since I have bought a road pedal / shoe system, and I have forgotten the differences between SPDs, Look pedals and all the others.    Can someone give a short comparison of each pedal system with pros and cons of each? My usage is/will be mainly city street cycling and then road cycling (recreation, not racing).  I'd prefer if people would give the pros and cons of each and not their opinion on the "pedal system war".  :) However, "I use them because I find XYZ and it helps in ABC ways" is very useful. I am trying to (partially) reproduce the algorithm displayed in Figure 4 from   My problem is with the multi-line conditions present in the "IFs". I want them to be indented, but \Indp does not seem to take effect if a new line \\ is not present in the same line.  This is what I tried:  \begin{algorithm}[htb]     \SetAlgoNoLine%     \SetNoFillComment     \SetKwIF{If}{ElseIf}{Else}{if}{:}{elif}{else:}{}%      \If{(SYNs from both hosts and \\             \Indp SYNs in earliest NFs of both hosts and \\             \Indp hosts' earliest packets differ)}{         Host with earliest NF is originator      }     \If{(SYN only from one host and \\         \Indp SYN is in connection's earliest NF)}{     Host is originator     }     \If{one host uses port 20 (ftp-data)}{     Host is originator     }     \If{only one host uses a well-known port}{     Host is responder     }     \If{start of hosts' first packets differ}{     Host with earliest NF is originator     }     Arbitrarily choose originator     \caption{Pseudo code for determining connection server side}     \label{algo:serverside2} \end{algorithm}   For instance, the line "SYNs in earliest NFs of both hosts and" is indented correctly, but "hosts' earliest packets differ" is not. XML attribute vs XML element At work we are being asked to create XML files to pass data to another offline application that will then create a second XML file to pass back in order to update some of our data. During the process we have been discussing with the team of the other application about the structure of the XML file.   The sample I came up with is essentially something like:  &lt;INVENTORY&gt;    &lt;ITEM serialNumber="something" location="something" barcode="something"&gt;       &lt;TYPE modelNumber="something" vendor="something"/&gt;     &lt;/ITEM&gt; &lt;/INVENTORY&gt;   The other team said that this was not industry standard and that attributes should only be used for meta data. They suggested:  &lt;INVENTORY&gt;    &lt;ITEM&gt;       &lt;SERIALNUMBER&gt;something&lt;/SERIALNUMBER&gt;       &lt;LOCATION&gt;something&lt;/LOCATION&gt;       &lt;BARCODE&gt;something&lt;/BARCODE&gt;       &lt;TYPE&gt;          &lt;MODELNUMBER&gt;something&lt;/MODELNUMBER&gt;          &lt;VENDOR&gt;something&lt;/VENDOR&gt;       &lt;/TYPE&gt;    &lt;/ITEM&gt; &lt;/INVENTORY&gt;   The reason I suggested the first is that the size of the file created is much smaller. There will be roughly 80000 items that will be in the file during transfer. Their suggestion in reality turns out to be three times larger than the one I suggested. I searched for the mysterious "Industry Standard" that was mentioned, but the closest I could find was that XML attributes should only be used for meta data, but said the debate was about what was actually meta data.  After the long winded explanation (sorry) how do you determine what is meta data, and when designing the structure of an XML document how should you decide when to use an attribute or an element? limsup of intersection of events as a subset of intersection of limsups Linux, find all files matching pattern and delete Looking to find all files (recursively) which have an underscore in their file name and then delete them via command line. I am going to install Ubuntu to dual boot with Windows 8 soon, but one of my friends warned me that partitioning my hard drive will slow my computer down. I run on a 500GB HDD and 8GB of ram if that helps at all. Is that true if I do it? Why can't I have a non-integral static const member in a class? I noticed C++ will not compile the following:  class No_Good {   static double const d = 1.0; };   However it will happily allow a variation where the double is changed to an int, unsigned, or any integral type:  class Happy_Times {   static unsigned const u = 1; };   My solution was to alter it to read:  class Now_Good {   static double d() { return 1.0; } };   and figure that the compiler will be smart enough to inline where necessary... but it left me curious.  Why would the C++ designer(s) allow me to static const an int or unsigned, but not a double?  Edit: I am using visual studio 7.1 (.net 2003) on Windows XP.  Edit2:  Question has been answered, but for completion, the error I was seeing:  error C2864: 'd' : only const static integral data members can be initialized inside a class or struct Reset select value to default Where should the period be put when an entire sentence is quoted at the end of a sentence? As in the following:     A common proverb is: “A rolling stone gathers no moss.”        A common proverb is: “A rolling stone gathers no moss”.        A common proverb is: “A rolling stone gathers no moss.”.   If the proverb weren't a full sentence, I would immediately put a single period after the quote, at the very end.  But if it is a sentence, which period should be removed so that there is only one period? Two periods look absolutely dorky. Prime number expressible as $2^2+3^3+5^5+7^7+11^{11}+\cdots$