I saw this problem in a math competition and I have no idea how to solve it:  $$\sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots }}}}$$
(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: .
How can I extract text from images?  I am not talking about scanned files, but garden variety images, such as when you take a high-def picture of a blackboard at class, and it is nicely handwritten; or when you photograph a page from a recipe book and want the recipe in text format.  Any free and open software for that?  I tried tesseract, and the results were awful. Coding for an ordered covariate I am performing ordinal regression, I have 5 response categories and several predictors both continuous and categorical. I would like to add a predictor which is categorical but ordered (1, 2, 3, 4). I don't think it would be appropriate to apply the usual dummy coding for unordered categorical predictors, but when I searched for how to code this I did not find much information.  In  "linear coding" or "assuming linearity of the predictor effect" is mentioned, but without further details.  Does it mean I just use my ordered values as they are, i.e. use them as a continuous variable? I wanted to get a 12 core CPU (Threadripper 1920X) and I wanted to know the max amount of cores Blender can use? Is $0\,\mathrm{m}$ dimensionless? Is $0 \, \mathrm m = 0 \, \mathrm s = 0 \,\mathrm {kg} = 0$? How do we define $[0 \, \mathrm m]$?  I once was given an assignment where I was asked to deduce and write down some physical quantity. It turned out that this quantity was $0$ in one unit or another, so I decided to drop the physical units because I figured that it didn't matter anyhow.  But I got to thinking:  If $0 \, \mathrm m = 0$, then we could add units of different dimensions (an operation not normally defined) like so: $5 + 0 \,\mathrm m = 5$.   Does $[ \cdot ]$ map $0 \, \mathrm m$ to both $0$ and $L$? Is there an established regime of thinking about physical quantities as an abstract algebra? Perhaps as a vector space where formal sums such as $5 \, \mathrm m + 10 \, \mathrm s$ are allowed? Would it be improper to drop the units of measurement of $0 \,\mathrm m$ like this in an academic paper? PostgreSQL Shapefile import failed I am very new to PostGIS/PostgreSQL. In fact, I downloaded it because of the answers I got to this question:   I installed both PostgreSQL and PostGIS. But I am not able to import a shapefile to the database? It gives an error, "Shapefile import failed". What am I doing wrong? Why does the sum of the reciprocals of factorials converge to $e$? I've been asked by some schoolmates why we have $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ I couldn't say much besides that the $\Gamma$ function, analytic continuation of the factorial, is defined with an integral involving $e$. Then I also know that actually $$ \sum_{n=0}^\infty \frac{x^n}{n!}=e^x.$$ Is there a reason for these facts?  P.S. I added the tag "intuition", please remove it if you think it is not pertinent. Bmp to jpg/png in C# From time to time I come up with ideas of actions I'd like to have available in the context menu of Nautilus. How do I add them there? Is there something like a Thunar UCA plugin for Nautilus? Disabling middle mouse button Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $ Easy way to model spiral engraving (knurling) that is usually used on metal for better grip? What is the easiest way to model this diamond grip pattern into a cylinder? Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation Find sum of the series $$\sum_{n=1}^{\infty}\frac{x^{3n}}{\left(3n\right)!}$$ using differentiation. So far I found that $$S(x)+1=S'''(x)$$ but it does not help. Then I tried different interesting ideas  like $$S(x)+S'(x)+S''(x)=e^x-1\,.$$Maybe if I get the third equation it will allow me to construct a kind of differential equation. Then, by solving it, obtain $S(x)$. Does the intersection of two finite index subgroups have finite index? It's really hard to get started on Stack Overflow I have been on Stack Overflow for some time.  It is an AMAZING resource and don't know if I would've got through the past 18 months without it.  I love it!  Problem is that I have "1" reputation.  All of my questions have already been asked and answered.  My "new user restrictions" mean: - I can't vote up answers that were helpful. - I can't comment on answers.  I sometimes have a question to ask one of the users that answered someone else's question but I can't.  This is hard!  How do I get a break?  Do I have to troll for a couple of days/weeks/months to answer enough questions in the hope someone votes some of them up just so I can do some basic functions?  Perhaps the number of site visits and length of membership could count for something? Let $x,y,z>0$ and $x+y+z=1$, then find the least value of ${{x}\over {2-x}}+{{y}\over {2-y}}+{{z}\over {2-z}}$ Let $x,y,z&gt;0$ and $x+y+z=1$, then find the least value of  $${{x}\over {2-x}}+{{y}\over {2-y}}+{{z}\over {2-z}}$$  I tried various ways of rearranging and using AM > GM inequality. But I couldn't get it. I am not good at inequalities. Please help me.  I wrote $x$ as $1-(y+z)$ and I took $x+y$ as $a$ and the others as $b$ and $c$. And I am trying. There are many lengths in LaTeX. For instance, \enskip, \enspace, \quad, \parskip, \smallskip, ...  Some of them are mostly used for vertical spacing, some others are mostly used for horizontal spacing, and perhaps some of them are used in both cases. I wonder whether there is a list that indicates the meaning and intended use of each length in LaTeX.  By "intended use" I mean not only whether the length is vertical or horizontal, but in which circumstances it is a good practice to use that specific length. Prove：$x^d-1 \mid x^n-1$ iff $d \mid n$. Consider the polynomial ring $F\left[x\right]$ over a field $F$. Let $d$ and $n$ be two nonnegative integers. Prove：$x^d-1 \mid x^n-1$ iff $d \mid n$.  my tries:  necessity, Let $n=d t+r$, $0\le r&lt;d$ since $x^d-1 \mid x^n-1$, so, $x^n-1=\left(x^d-1\right)\left(x^{\text{dt}+r-d}+\dots+1\right)$... so,,, to prove $r=0$? I don't know, and I can't go on. How to do it. How bad for TeX is omitting braces {}, even if the result is the same? Sometimes writing braces and omitting them produces the same result:  \tilde{\psi}  \mathrm{e}^{\mathrm{i} \pi}   vs  \tilde \psi   \mathrm e^{\mathrm i \pi}   Or (taken from Werner's comment below)   \frac{1}{2} vs \frac12   If that's the case, how bad for the quality of my code is omitting them? Of course, if one compiles a pair of formulas, there is no difference. But, since I ignore how TeX works, my point is: if one writes long codes with these habits,   doesn't TeX go somehow crazy because I'm not writing as it expects?   For instance, does it have any effect on how fast are long documents compiled? Any reason to keep them? I'm working under 12.04 and suppose I have a Firefox windows active (or in foreground) with English as input language and I need to type a document in other language using some text editor.   With the text editor in foreground (or active) and the input language set to a non-English one, when I bring Firefox in foreground (or making it active) the input language remains set to the non-English and the language flag does not switch to English (as it would be expected, since I do not alter the language during the whole Firefox session).  Because of this, I have to make extra moves and change the input language manually every time I switch from the text editor to Firefox and back to text editor. This was not happening with 10.04, and each application windows had the corresponding input language set to its default or previous session every time I was bringing it to the foreground!   How will I make 12.04 to behave the same way? How to print original variable's name in Python after it was returned from a function? I have enum and use the variables like myEnum.SomeNameA, myEnum.SomeNameB, etc.  When I return one of these variables from a function, can I print their names (such as myEnum.SomeNameA) instead of the value they returned?