Suppose we have the following two identities:    $\displaystyle \sum_{k=0}^{n} \binom{n}{k} = 2^n$ $\displaystyle \sum_{k=0}^{n} (-1)^{k} \binom{n}{k} = 0$   The first says that the number of subsets of an $n$-set is $2^n$. The second says that the number of subsets of even size equals the number of subsets of odd size (of an $n$-set). Thus there are $2^{n-1}$ subsets of even length and $2^{n-1}$ subsets of odd length?  To combinatorially prove the second identity, let $A$ be a $k$-subset of $[n]$. Then note whether $k$ is odd or even?
Show that $$\sum_{k=0}^n(-1)^k\binom nk=0$$ So for odd $n$ we have an even number of terms. So $\binom nk=\binom n{n-k}$ which have opposite signs. Thus the sum is 0.  For even $n$ we have that $$\sum_{k=0}^n(-1)^k\binom nk= \binom n0+\sum_{k=1}^{n-1}(-1)^k\binom nk+\binom nn$$ Now $$\sum_{k=1}^{n-1}(-1)^k\binom nk= \sum_{k=1}^{n-1}(-1)^k\left[\binom{n-1}k+\binom{n-1}{k-1}\right]$$ What would that sum be in the square brackets?
I am looking for an animated series, shown in the 90s. I remember an episode where the boy went into VR and fought a medusa-like monster.  It may be that this was the premise for the whole series: That the children heroes go into VR every time to fight the evil guys.   I am not sure if the whole series were computer-animated but surely there were such elements together with classic animation.  The style of the series was quite dark and cool for teenage boys. Is there a place to dispute duplicate markings? I just installed my first Ubuntu Server (12.04) instance and immediately ran into the following problems:   Running apt-get install or update fails (see below) Running ifconfig produces peculiar IP addresses starting with 10.x.x.x and my router does not see the server at all in its "Attached Devices" page (its a consumer grade Netgear)   When I run apt-get I get 50+ of the following errors:  Failed to fetch &lt;some-http-url&gt;. Temporary failure resolving &lt;blah&gt;.ubuntu.com   I read up on this error for 12.04, and found some similar posts on this forum and on server fault that recommended going into /etc/resolv.conf and adding an entry:  nameserver 8.8.8.8   To my surprise, I found that /etc/resolv.conf was a symlink pointing to somewhere else  and I had to delete it first, then create a new one with the entry indicated above.  I restarted the server and tried running apt-get again, same results.  How do I start diagnosing the problem (I'm a relatively new Linux/Ubuntu user)?  Additional details that may help:   This is actually a virtual machine running as a VirtualBox guest OS The physical host that is running this VM is my laptop which has a wireless connection; I'm wondering if (somehow) the laptop is getting the network wirelessly, but perhaps either the VBox VM and/or the Ubuntu server OS isn't configured for wireless and so nothing is "getting through"?   Thanks in advance for any pointers!  Edit:  Here is a snapshot of my VM's Network settings (only Adapter 1 has any information populating its fields, so I assume my VM has only 1 adapter):    Here is the output when I run ifconfig from the terminal:    And here is the output when I run route from the terminal:    Running ping google.com: Did the date of Frodo's rescue out of the fire of Mordor have any particular meaning in The Lord of the Rings? Did the date of Frodo's rescue out of the fire of Mordor have any particular meaning in The Lord of the Rings?  We read in The Return of the King (Book 6 Chapter 4) that Gandalf states to Frodo and Sam:     The fourteenth of the New Year, or if you like, the eighth day of April in the Shire reckoning. But in Gondor the New Year will always now begin on the twenty-fifth of March when Sauron fell, and when you were brought out of the fire to the King.   Did Tolkien have any particular reason for choosing this date for Sauron's fall and Frodo's rescue out of the fire of Mordor, either allegorically or otherwise?  We are aware that Tolkien was not a fan of , but preferred the term applicability instead. If $a_k\geq0$ is a decreasing sequence converging to $0$, do we have $\lim_{n\to\infty}\frac{\sum_{k=1}^n a_k}n = 0$? What is the best way to handle a lot of misused tags? So today I noticed a question that had the  tag. As the question was not about JSFiddle, and only included a link to the site, I thought somebody has just created this for the sake of it. So I decided the best action would be to remove it, and try to stop it from getting out of control.  But then I searched (clicked the tag) for other questions that used it and I was amayonaised to see over 500 questions had the very same tag! Most of which were incorrectly tagged. So I cleaned up a couple and then quickly got bored.  So I am wondering, aside from going through them all by myself, is there a more efficient way?  I think it would be a great idea if there was some feature (probably an admin feature) that allowed a new tag queue to be added to the review list. I think this could work well as follows:   Somebody brings up the problematic tag in question (e.g. Me on here) An admin reviews my concern, and decides that the tag is very misused That admin adds the tag for review in a (temporary) new review queue. All users that can review can then go through existing tags, and vote for "No action" or "Remove tag" The world becomes a slightly better place   ...or not?    By the way, I tried to add some new tags to this post but I don't have the rep. So if someone could create them for me that would be swell: "musefan", "today", "sunny-spells", "pointless-tags" My wife wants a 4th baby, but I don't What are your favorite English language tools? We have a system hosting many websites for our customers, and inside that system there is a method that non-live customers can view their sites before we turn them on.  Say the link is something like this: ourbigcompany.com/customer/domain=thisisanewsiteurl  Those links are not linked to anywhere outside a secure login - they are only sent to the customer via email. They are publicly viewable, as they have to be, but that's not the real problem.  The real problem is that somehow Bing is getting hold of them and trying to crawl the sites.   I know how to stop the crawling, but that would be like treating the symptoms without fixing the problem.  We log the traffic and there is no referrer - so that is not helpful.  If I change the querystring value for a site, Bing has it within hours. I need to figure out where Bing is getting the links from so that I can close what is obviously a security hole, but I am not sure how.   Any ideas on how to figure that out? Upgrading from Yosemite to El Capitan, but not Sierra We have an older iMac (Model A1311, Mid 2011). It has been running Yosemite. We have run into some of the WIFI issues associated with Yosemite.   Is it possible to upgrade to El Capitan but not Sierra?   El Cap has been out longer. Its my wife's computer, its older, and I really don't want to be debugging Sierra until a few  more "dot" releases come out. $$\frac{1}{n+1}&lt; \log(1+ 1/n)$$  Any ideas? I tried estimating the difference between $1/n$ and the logarithm and comparing with $1/n-1/(n+1)$ but I miserably failed. I have windows 7 in Arabic. Now I dowloaded QGIS 2.14.3-2 in French.  But when I installed it shows Arabic menu/tools...  How can I change the menu language to French? Distribution of the digits of Pi Using (or not) the word "to" in a sentence ending with "be able" On Windows 7, is there a command prompt line that can start cmd as an administrator? On Linux, we can use  sudo bash   to run a shell as the superuser.  On Windows 7, we can use     Start menu -> (in search box, type) cmd -> Right click on search result and choose Run as Administrator   to run a command prompt as an administrator. Is there also a command prompt line that is like Linux's sudo or su so as to start a shell as an administrator? What does 1x1 convolution mean in a neural network? I am currently doing the Udacity Deep Learning Tutorial. In Lesson 3, they talk about a 1x1 convolution. This 1x1 convolution is used in Google Inception Module. I'm having trouble understanding what is a 1x1 convolution.  I have also seen  by Yann Lecun.  Could someone kindly explain this to me? When typing &lt; or &gt; in LaTeX and compiling with pdflatex, the less than and greater than symbols appear at upside down exclamation points. I'm not in math mode. How big is "too big" for an airline seat? At what point will you be required to buy two tickets? As I said in another question, I'm bringing my parents to visit me here in Sweden next year.  My mother is around 280-300 pounds and 5'6" (130-135kg, 1.67m). I am concerned that she might be asked at the airport to buy a second seat. If it's likely I would rather just buy the second seat in advance, because I don't want her to be embarassed/humiliated by some random employee. As well, I won't be there, they will be travelling alone and aren't used to dealing with airports and they won't know what their rights are.  In terms of how it will affect other passengers, it's not important, she will get a window seat and my dad will sit in the middle seat, so I don't see it affecting anybody.   She insists it's fine, "the last time I flew I fit easily!" but that was 20 years ago when she was half the size.   They will be flying BA or SAS, definitely not Ryanair.  So - where is the cutoff? Is a 300 pound 5'6" woman going to be too big to fit in a single airline seat? Or am I worrying about nothing? Is it possible to browse a chatroom under invisible mode? E.g.  I know that I can browse the chat when I'm not logged on. But what if I am logged on and want to spare the inconvenience of logging off/multiple browsers? How do I call a Windows shell command using VB6?