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Calculate sums of inverses of binomial coefficients How to calculate the sum of sequence $$\frac{1}{\binom{n}{1}}+\frac{1}{\binom{n}{2}}+\frac{1}{\binom{n}{3}}+\cdots+\frac{1}{\binom{n}{n}}=？$$ How about its limit? How did Gandalf elude and puzzle Sauron? In the History of Middle-Earth book collection Myths Transformed we learn a lot about how Sauron perceived the Istari and what he knew of them.     Sauron had, in fact been very like Saruman, and so understood him quickly and could guess what he would be likely to think and do, even without the aid of the palantiri or of spies; whereas Gandalf eluded and puzzled him.   (Home X: Myths Transformed)     Gandalf he did not understand. But certainly he had already become evil, and therefore stupid, enough to imagine that his different behaviour was due simply to weaker intelligence and lack of firm masterful purpose. He was only rather a cleverer Radagast &mdash; cleverer, because it is more profitable (more productive of power) to become absorbed in the study of people than of animals.   (ibid)  My question now is how does Gandalf elude and puzzle Sauron so much? We know he found Saruman simple to guess down to having similar minds (maybe down to both being maia of Aule?), we also know he must know a fair bit of Radagast as he refers to Gandalf as merely a smarter version of Radagast who was known to have abandoned his task which the Istari were given.   While Sauron went face to face with Saruman via the palantir, it is surprising that he finds Gandalf basically strange, even though Gandalf in reality does the most to circumvent Sauron out of the Istari in the north-west.   Gandalf enters Dol Guldur which Sauron flees. Gandalf enters Dol Guldur again in secret and claims he only just escaped with his life,  which suggests Sauron may have ended up noticing him. Gandalf pushes White Council to attack Dol Guldur which Sauron foresees. Gandalf fought with all nine of Sauron's greatest servants at Weathertop and played a part in destroying their physical bodies temporarily. Gandalf leads the Fellowship from Rivendell, which Sauron also knows. Gandalf strives with the Dark Tower when Frodo puts the ring on at Amon Hen. Gandalf is present at the Battle of the Black Gate, in which the mouth of Sauron declares they know all about Gandalf.   As I've pointed out, Gandalf is clearly sent to contest Sauron. But I don't understand why Sauron does not see this; throughout the Third Age, well, since the coming of the Istari Gandalf has played a role in almost every contest against Sauron.  So how did he elude and puzzle him? Is there a software in Ubuntu that can analyse the cross references (theorems, lemmas, equations, etc.) in a tex project? There are some software to analyse the cross references among codes (Java or C++). I used to use a software for code analysis under Windows many years before. But I can't remember the name. On linux, an example might be LXR Cross Referencer.   Is there a similar tool that can analyse the latex files to find out the cross references of theorems, lemmas, propositions and equations? What is ISCAS85 benchmark? What is ISCAS85 benchmark? I find it in a , is there any simulator for it? I read the answers in this question, but they still leave my question open:   I didn't find a way in the Google Webmaster Tools to invoke a re-download of my robots.txt.  Through some error, my robots.txt was replaced with:  User-agent: * Disallow: /   And now all my content was removed from Google search results.  Obviously, I'm interested in correcting this as soon as possible. I already replaced the robots.txt, but I can't find a way to make Google update the cached version. mod [= remainder] operation (and relation), name and meaning I am trying to write the Euclidean algorithm in the following way:  $A = \lfloor A \div B \rfloor \times B + (\text{remainder of}) \: A \div B $   Now is there any symbol I can use  to say "remainder of A $\div$ B"? I know that in the C programming language there is the operator % for modulus; is that a valid symbol in maths? Can I write A % B? Or is there some other way? Root filling up although it should have some space This has been perplexing me for some time. I'm on Ubuntu 11.04. My root partition is filling up and I can't figure out what's causing it. Even stranger, when I run df -h, this is the output:  Filesystem            Size  Used Avail Use% Mounted on /dev/sda2             9.2G  8.8G     0 100% /   As you see, there should be about 400 MB free, but Avail shows 0. What could be causing this? I've even uninstalled some programs to free the space and it fills up again. Suggestions on how to find out which, if any, files are filling up the space are also welcome. I'm from Tunisia and I got a Schengen visa from the Austrian Embassy for  28 days, the “duration of stay”, and the number of entries is 1. But the validity of the visa is from 02/08/2015 to 13/09/2015.  Do I have to stay only 28 days, for example from 02/08/2015 to 29/08/2015 or can I stay there until 12/09/2015 ?  And if I have got to stay only 28 days, then can I choose the days of stay, that's mean can I stay in Austria for example from 09/08/2015 to 05/09/2015? I was working on a worksheet for class and came across this problem:      Let $\varphi:R\to S$ be an onto ring homomorphism with the property that every ideal of $R$ is principal. Show that the same must be true for $S$.    I honestly do not know how to do this one! Why isn't "Column ordering" available in my list settings? I want to change the column order visible when viewing or editing a list item. I have accessed the "Column ordering" interface via the list settings page on other lists in the past, but it isn't visible in this list's settings. Repeatedly rolling a six sided die four times and summing the highest three results gives you a distribution with what mean and standard deviation? Repeatedly rolling a six sided die four times and summing the highest three results gives you a distribution with what mean and standard deviation?  I've only taken AP statistics, but I would like to learn how to do this. I'm hoping there's something in the same conceptual space as the old VB6 IsNumeric() function? Can I install a ROM made for a different device? How to read high voltages on microcontroller? The abstract environment changes the \linewidth for the whole document In the article class, it seems that using the abstract environment has an effect to the rest of the document. More precisely, it seems that it redefines the \linewidth, which equals \textwidth without abstract, is smaller with the abstract. My feeling is that \linewidth is redefined for the abstract environment, but the value is not reseted at the end of the abstract.  Here is a minimal example:  \documentclass{article} \begin{document} \abstract{This is an abstract. This is an abstract.}\bigskip  This is text. This is text. This is text. This is text. This is text. This is text.\\  \rule{\textwidth}{3em} \rule{\linewidth}{3em} \end{document}   The output is not the same if you comment the \abstract line or not.  Do you have the same problem as I do when you compile (if not, I will upload some PDF files to show)? Is this a (known) bug in LaTeX2e? How can we circumvent the problem?  Note: I compile with the pdflatex command, and I use pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian). $\sum_{i=1}^n(a_i)\sum_{i=1}^n(\frac{1}{a_i})\ge n^2$ is what I want to prove, where $a_i&gt;0$ and I need to showwhen equality holds.  I have tried using Cauchy-Schwarz, where inside the sums I multiply each value by 1 to do it but it doesn't work out.  I have also done  $\sum_{i=1}^n(a_i)\sum_{i=1}^n(\frac{1}{a_i})=\sum_{i=1}^n \sum_{i=1}^n(\frac{a_i}{a_i})=\sum_{i=1}^n\sum_{i=1}^n(1)=\sum_{i=1}^n(n)=n^2$, but this proves an equality.  Am I not allowed to do the first step I did in my work? Why shouldn't the denominator of the covariance estimator be n-2 rather than n-1? Trying to run a Ubuntu server for the first time... But everything is in a terminal. Is there a way to switch to a GUI? When to use single quotes, double quotes, and backticks in MySQL Why do the humans become sleepy after meals?