Let : $$f(x)+f\left(\frac{1-x}{x} \right)=x$$ then: $$f(x)=?$$   My Try :  $x\to  \dfrac{1-x}{x}$  \begin{align}f\left(\dfrac{1-x}{x}\right)+f\left(\frac{1-\dfrac{1-x}{x}}{\dfrac{1-x}{x}} \right)&amp;=\dfrac{1-x}{x}\\ f\left(\dfrac{1-x}{x}\right)+f\left(\dfrac{2x-1}{x^2} \right)&amp;=\dfrac{1-x}{x}\end{align}  And $u=\dfrac{1-x}{x}$  $$f(u)+f\left(\dfrac{2x-1}{x^2} \right)=u$$  Now what ?
What function satisfies $ f(x)+f\left(\frac{1-x}x\right)=x$ ?
Fourier series identity I need to prove that $\dfrac{a \sin(bx)}{1 - 2a \cos(bx) + a^2} = \sum_{n=1}^\infty a^n \sin(nbx)$ where $|a| &lt; 1$.  It seems that this can be proved by using Euler's formula identities for $\cos(bx)$ and $\sin(bx)$ and substituting $z = e^{ibx}$. From that, I get $$ \dfrac{a \sin(bx)}{1 - 2a \cos(bx) + a^2} = \dfrac{a (z - 1/z)}{2i(1 - a(z + 1/z) + a^2)}$$ $$= \dfrac{a (z^2 -1)}{2i(z - a(z^2 +1) + za^2)}$$ $$= \dfrac{a (z^2 -1)}{-2ai(z^2 - z(1-a^2)/a + 1)}$$  I don't understand what I am supposed to do next. A few months ago I set full disk encryption during installation of Ubuntu 17.10. Now, I've decided to upgrade. The upgrade ran to the end without any problems. However, after reboot I am not able to log into my encrypted disk.  Where can be the problem? I am 100% sure I press the "right keys" on the keyboard, but technically I don't know what I am writing because of "*" characters and maybe after upgrade my keyboard layout changed. I use some characters which might be somewhere else on the keyboard. What language is default after upgrade?   By the way, I've already tried Caps-lock, but still no luck.  Help will be really appreciated. I am not a fan of installing my system and all backups over and over again. How to make the numbering of subsections independent from the numbering of sections? I am writing a constitution for a society.  I would like to have the subsections numbered independently of the way that the sections are numbered, i.e:  Section 1     Subsection 1     Subsection 2 Section 2     Subsection 3   How can this be done? Are Harry Potter and Ginny Weasley Distant Relatives? Is it ever revealed what Crookshanks is? When Hermione first got her pet thing Crookshanks, there was a mention that it looked kinda like a cat, but wasn't.  Was it ever revealed what Crookshanks really was? Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime. Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime.  Can someone help me with this? Let $L$ be a finitely generated free module over a commutative ring $A$. Set $n=\operatorname{rank} L$. Let $x_1,\dots,x_m$ be generators of $L$. Then $m \ge n$? If $m = n$, then is $x_1,\dots,x_m$ a basis of $L$? Consider the surface $S$ in $\mathbb{R}^3$ given by the graph of $f(x, y) = xy^2 − 2y^2 + e^x$ . Calculate the intersection point of the tangent planes to $S$ above $(x, y) = (0, 0), (0, 1)$, and $(0, 2)$.  Can someone guide me through solving this problem? I am confused on where to start because we didn't cover this in class. Thanks so much Continuity of $\sin(x)/x$ Is $\sin(x)/x$ continuous at $x=0$,(perfectly, not tends to)? The function has right hand and left hand limit same and equals 1 but what happens at perfect zero?it should be undefined. Is it? Does a photon exert a gravitational pull? I know a photon has zero rest mass, but it does have plenty of energy. Since energy and mass are equivalent does this mean that a photon (or more practically, a light beam) exerts a gravitational pull on other objects? If so, does it depend on the frequency of the photon? On any SE post one can use the backtick symbol ("`") in order to mark some text as being a piece of code, or computer output. For example: void main().  Is there a way to make use of "`" the same way in a LaTeX document? Will grammar errors become correct after enough people use them for enough time? First let me state the obvious—based on my own experience—that hordes of people are confused about certain basic grammar principles. For example, I so often see mistakes in choosing the pronoun to use with direct objects, especially when introduced by and with another person, as in "Joe talked about the book with Anna and [I/me]." Another one is the near 100% usage of "lay" instead of "lie" (and "laid" instead of "lay") nowadays, even in published sci-fi books that most certainly had professional editors proofread them.  Do you see these kinds of changes as predictors of the future? Will they become correct after enough people have used them long enough?  Both of these problems drive me bonkers, and I fight my own little battle against them when possible.  I know that dictionaries are descriptive rather than prescriptive. They're technically history books. But grammar and style books seem less so, to me. Those seem as though they ought to be prescriptive in order to reduce the erosion of structured and meaningful language.  In this tension between "what people are doing" and "what people should be doing," are we doomed to forever bear these errors just because too many people don't know the proper way? Is it worth fighting against them? Do I capitulate to the people who are quick to accept any mass public usage as a fait accompli?  Update  I don't imagine for a second that my own writing is error free. The difference with me is that when I do learn I have something wrong in my mind, I immediately change once I clearly understand it.  I also am not imagining that language is static or ignorant that today's correctness is all too often yesterday's error. The whole point of my question here, then, is: is there value in slowing the change, and if so, how is that done and how effectively?  Another way to look at what I'm trying to ask is that in a way I am hoping to define the limits of pedantry. If being pedantic is slavish adherence to outdated rules in the face of actual and foregone changed reality, then when do we conclude that a change is a foregone conclusion?  Poking fun at my own errors to highlight supposed pedantry on my part is to miss everything I am trying to say.    Update 2  It may not be an easy question, but since language change actually does or does not happen, and every person acts with more or less intention in regards to language change, there must be an answer. What I wanted to explore is the value in efforts to teach the "proper way" vs. the opposite end of the spectrum where any new usage is not just accepted but welcome or even sought out. You know, I'm reminded of something: the conservative/liberal scale:     |------+------+------+------+------+------| radical              moderate             radical liberal                                 conservative   The funny thing about those at the ends of the spectrum is that they both want things to change. The radical liberal wants language to change to something new quickly, just for the sake of newness or evolution or some other not-necessarily-realistic ideal. Similarly, the radical conservative wants things to change just as much, but back to the way it used to be, just for the sake of sameness or continuity or some other just-as-not-necessarily-realistic ideal.  I'm not particularly asking where we should be on this scale as that's purely subjective. I'm trying to ask about something purely practical and real-world: what is actually effective? Is language change inevitable, and how fast? Are efforts to reduce its rapid morphing either effective or worthwhile in any measure? All these things considered, what position with respect to language evolution is livable, practical, and sensible?  Feel free to edit my grammar. No comment necessary.  Update 3  Something that may be of interest to both my supporters and detractors: . I enjoyed it. I don't disagree with him, but I am not sure I am wholehearted in this lack of disagreement. How to use C++ 11 features in the latest version of g++ Newbie here. I just got an error running from the terminal a C++ program I wrote: error: ‘stoi’ is not a member of ‘std’. They told me the compiler is too old.  I'm using Ubuntu 14.04.  My g++ version is 4.8.4.  How do I upgrade? I know that the Naruto movies are outside the timeline in the sense that they don't affect whatever happens after them. However, they must fall somewhere in the Naruto timeline — based on who are in the movies and how the characters look like, etc. For example, if Sasuke isn't in the movie, it should take place after he left. For each of the movies, approximately what timeframe (between episodes/arcs) are they set?  The movies:   Ninja Clash in the Land of Snow Legend of the Stone of Gelel Guardians of the Crescent Moon Kingdom Naruto Shippuden the Movie Bonds The Will of Fire The Lost Tower Blood Prison Road to Ninja The Last - Naruto the Movie Boruto: Naruto The Movie Black screen after suspend-after moving to the nvidia official drivers (not priority) So, when I used the priority drivers, the suspend worked normal. But the official drivers works better, so I installed the official driver. Then suspend won't work.  Should I change any text/documants to make it work? Any solution will be aprreciated, thanks you all! Is there an "exists" function for jQuery? How can I check the existence of an element in jQuery?  The current code that I have is this:  if ($(selector).length &gt; 0) {     // Do something }   Is there a more elegant way to approach this? Perhaps a plugin or a function? Evaluate $\lim_{n\to \infty} \sum_{r=1}^n \frac {1}{2^r}\tan \left(\frac {1}{2^r}\right)$ Evaluate $$\lim_{n\to \infty} \sum_{r=1}^n \frac {1}{2^r}\tan \left(\frac {1}{2^r}\right)$$   I tried to create a telescoping sum but I couldn't. The last step I could reach was turning the limit into $$\lim_{n\to \infty} \sum_{r=1}^n \left(\frac {1}{2^r(1-\tan (2^{-r+1})} -\frac {1}{2^r(1+\tan (2^{-r+1})}\right) $$  But couldn't proceed further. Also I thought about Riemann sums but it was a pure dead end.  Any help would be greatly appreciated I'm a bachelor student majoring in math, and pretty interested in physics. I would like a book to study for classical mechanics, that will prepare me to work through . What books would be good for a beginner? Why aren't search engines indexing my content? Notice these sentences:   Shops are open late in summer.   Summer is a traditionally viewed as a slow season for games.   At the summer solstice, the days are longest and the nights are shortest.   it's time to start thinking about what 2016's Song of the Summer will be.    In English class, they told us that "the" is used for the things that are known.  One example for this was a phrase like "The man who killed his wife".  So when we are speaking of that man, we know what he did. Somehow we know something more than nothing about him. He's not just every man in the world.     So Now I'm confused about why "the" is not used in the first and second sentences. Can anyone please explain it for me?