Evaluate: $$S=1+ \frac{1}{2+ \dfrac{1}{4+ \dfrac{1}{8+ \dfrac{1}{16 +\cdots} }} }$$ What have I done, so far? Nothing, couldn't think of a way to solve it. But I know it is around 1.46. I want to solve it myself, so hint on how to solve will be sufficent.
Say you have the infinitely continued fraction: $$x+\cfrac{1}{x^2+\cfrac{1}{x^3+\cfrac{1}{x^4+\ddots}}}$$ When $x=1$, you can see that it's $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}$$ which converges upon the golden ratio $\phi = 1.61803398875...$. What if I were to plug in $x=2$? $$2+\cfrac{1}{4+\cfrac{1}{8+\cfrac{1}{16+\ddots}}}$$ Using a caluclator, it seems to converge upon the irrational number: $2.24248109286...$ Is there any way I can represent this irrational number algebraically?
Searching for questions that have ONLY a given tag (i.e. add a range operator for tags) Can we make it more plain to recruiters that we're not open to relocation? The preference for being open to relocation is a checkbox, and if you don't check it, nothing about relocation shows up in your profile.  Evidently recruiters can't understand that a lack of "I am open to relocation" is the same thing as "I am not open to relocation", and so every single contact I've gotten through Careers is for a job that I would have to relocate for.  Can we make it more obvious to the recruiters (especially ones from Amazon) that we have not checked that box on purpose? Display only relevant reputation points If cooking meat kills bacteria, and bacteria are responsible for problems with eating meat which has been left out, then why is it dangerous to eat meat which has been left out at room temperature and then thoroughly cooked?   mentions that "Even if the bacteria is dead, toxins can remain if the food was out too long, causing problems".  However, there's no further detail given.  Are these toxins as dangerous as the bacteria themselves?  How long does meat have to be left out to accumulate a dangerous level of toxins and thus be dangerous even if thoroughly cooked?  Are these toxins the reason for the usual guideline of keeping meat unrefrigerated for a maximum of 2 hours?  update  The revelation, courtesy of Aaronut, that e. coli is actually dangerous because of its toxins — which cannot be denatured at temperatures which will leave meat in an edible state — has pretty much answered this question.  And also given me further incentive to stop eating meat altogether :)  Our discussion (see the comments on hobodave's answer) has progressed into the realm of microbiology.  Some highlights from my ongoing research:  .  These seem to be the reason for the importance of keeping meat at a high temperature for a period of time.  .  This also provides fascinating insight into how bacteria evolved immunity to antibiotics.  Fungi. How to correct for small p-value due to very large sample size I am running into a problem where an independent variable, which should have no predictive power on the dependent variable based on domain knowledge, comes out with very small p-value because the sample size is very large(~100,000). If I only use &lt; 5000 data points, then the p-value becomes large enough to support the prior that the variable is insignificant. However, I don't think tweaking the sample size to get to the desired conclusion is a good practice. Is there any procedure to adjust for small p-value simply due to huge sample size? Effective Refractive Index in Paper I am reading the paper, entitled "Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings" by Eric B. Grann, M. G. Moharam, and Drew A. Pommet.   In the paper they mention this formula:        Do you know how it can be derived and where it comes from ? How to capture all std outputs How to mount an internal HDD on start-up? I have a 500GB internal HDD where I store all my music and photos but it is annoying to manually mount it after a restart and I was wondering how to mount it automatically and make it available to Rhythimbox for playback.  Thanks! Suppose that $X_1,X_2,...,X_n$ is a random sample from a normal distribution, $X_i\sim N(\mu,9)$.  Find the UMVUE (uniformly minimum variance unbiased estimator) of $P(X\le c)$ where $c$ is a known constant.  Do this by finding the conditional distribution of $X_1$ given $\bar{X}={\bar{x}}$ and apply the Rao-Blackwell theorem with $T=u(X_1)$, where $u(x_1)=1$ if $x\le c$ and zero otherwise.  So clearly the parameter that we're trying to estimate given the problem description can be written as $\Phi\left( \frac{c-\mu}{3} \right)$ where $\Phi$ represents the CDF of the standard normal distribution.  However, I get stuck when I need to try and find the conditional distribution.  Without knowing the joint distribution or the other conditional distribution, how can I possibly find the conditional distribution of $X_1$ given $\bar{X}={\bar{x}}$? What are acceptable reasons for flagging as "Requires Moderator attention"? After reading , I feel I don't quite understand this function. What are the guidelines for flagging a question as &quot;Requires Moderator attention&quot;? I have two Date values. I need to find number of weeks between them. I want to do it Apex Class. Not looking for formula fields.  So far I tried:  Integer numberofweeks = (projectStartDate - opp.Project_Start_Date__c)/7;    I receive this error:     Date arithmetic expressions must use Integer or Long arguments Fastest way to convert string to integer in PHP How to make the Java.awt.Robot type unicode characters? (Is it possible?) We have a user provided string that may contain unicode characters, and we want the robot to type that string.  How do you convert a string into keyCodes that the robot will use? How do you do it so it is also java version independant (1.3 -> 1.6)?  What we have working for "ascii" chars is  //char c = nextChar(); //char c = 'a'; // this works, and so does 'A' char c = 'á'; // this doesn't, and neither does 'Ă' Robot robot = new Robot(); KeyStroke key = KeyStroke.getKeyStroke("pressed " + Character.toUpperCase(c) ); if( null != key ) {   // should only have to worry about case with standard characters   if (Character.isUpperCase(c))   {     robot.keyPress(KeyEvent.VK_SHIFT);   }    robot.keyPress(key.getKeyCode());   robot.keyRelease(key.getKeyCode());    if (Character.isUpperCase(c))   {     robot.keyRelease(KeyEvent.VK_SHIFT);   } } What does it mean for two objects to "touch"? Why do neural networks need so many training examples to perform? A human child at age 2 needs around 5 instances of a car to be able to identify it with reasonable accuracy regardless of color, make, etc. When my son was 2, he was able to identify trams and trains, even though he had seen just a few. Since he was usually confusing one with each other, apparently his neural network was not trained enough, but still.  What is it that artificial neural networks are missing that prevent them from being able to learn way quicker? Is transfer learning an answer? I think the single dashes (hyphens) should be placed between the numbers and units in this sentence:     ...in a cylinder of 3-inch radius and 24-inch length.   Is that correct and should hyphens be added to this second sentence?     ...making a trench nearly 15 feet wide and more than 30 feet long. The bikeshed problem and Stack Exchange In short, :     If you go before the Board of   Directors and ask for 1.5 billion   dollars to build a nuclear reactor, no   one will review or discuss the details   of the plant. They will assume that   experts have been over every inch of   the plans, and not want to look   foolish by asking a silly question.      However, if you ask the same group to   approve a 30 dollar expenditure for   lumber with which to build a bikeshed   (presumably a British term for the   smallest possible building) then be   prepared for a 45 minute discussion   about all aspects of the bikeshed,   including the color of the paint. The   explanation for this is that everyone   can grasp the scale of a bikeshed.   To transform this problem to Stack Overflow terms, questions that are trivial such as "How to convert a C string to a QString or "What is the difference between . and ->" get a lot more views, answers and votes than real questions of the form "How do I frob this widget" that bother real professionals. Fewer views mean less possible votes for good answers and hence less incentive for people to answer.  How can Stack Exchange solve this problem? There are many cards that can put multiple permanents onto the battlefield simultaneously, such as , , or .  How do these interact with the following types of abilities?  As [this permanent] enters the battlefield... You may have [this permanent] enter the battlefield as... [This permanent] enters the battlefield with... When [a permanent with certain characteristics] enters the battlefield...   Justification There have been many questions already about particular instances of this situation.  This question is intended to be a canonical question and answer to all questions such as:      Note to moderators: three of these five questions have already been marked as duplicates pointing to this one.  Can we do the same for the remaining two? This question led to a new package:      I want to reference to citations from other file. Similar to what xr package does to labels, but I want to apply it for the bibliography.  If I'm able to do \cite{someone} in the main file, and it creates let say the number [4], I want to do the same \cite{someone} in a second file, and also create the same number [4]. Similar to what xr does with the labels.  By the way, I tried to add the main.bbl file from the main file. However, that adds all the references to the end of the second file. I want to avoid this.  Is there any package that accomplish this? For Christmas my kiddo got the D&amp;D Starter Set and we've been going though it and the full rules we downloaded online. I'm having a hard time figuring out how the example listed in the rules gets to a final value of 14.  The example is a level 1 character with wisdom of 15 and a proficiency in perception.  I understand that you start with  10 (base) + 2 (character level) + 2 (wisdom 15) which gets you to 14 but then how does the "proficiency in perception" help or factor in?   What gets more confusing is on one of the pre-generated character sheets has a level 1 character with wisdom of 10 but a passive perception value of 10. If I follow the math from above shouldn't it be 10 (base) + 2 (char lev 1) + 0 (wisdom 10 = 0 bonus) for a final score of 12?  How to calculate passive perception?