What is the grandchild of a parents's siblings called?
What is the term for the grandson of my aunt?
How can I distribute a native executable for a Python program? Object-oriented shell for *nix Let $x_1,x_2,\ldots,x_n$ be the roots for $1+x+x^2+\ldots+x^n=0$. Find the value of   $$P(1)=\frac{1}{x_1-1}+\frac{1}{x_2-1}+\ldots+\frac{1}{x_n-1}$$   Source: IME entrance exam (Military Institute of Engineering, Brazil), date not provided (possibly from the 1950s)  My attempt: Developing expression $P(1)$, replacing the 1 by $x$, follows $$P(x)=\frac{(x_2-x)\cdots (x_n-x)+\ldots+(x_1-x)\cdots (x_{n-1}-x)}{(x_1-x)(x_2-x)\cdots (x_n-x)}$$    As $x_1,x_2,\ldots,x_n$ are the roots, it must be true that   $$Q(x)=(x-x_1)\cdots(x-x_n)=1+x+x^2+\ldots+x^n$$ and  $$Q(1)=(1-x_1)\cdots(1-x_n)=n+1$$ Therefore the denominator of $P(1)$ is  $$(-1)^{n} (n+1).$$ But I could not find a way to simplify the numerator.  Another fact that is probably useful is that  $$1+x^{n+1}=(1-x)(x^n+x^{n-1}+\ldots+x+1)$$ with roots that are 1 in addition of the given roots $x_1,x_2,\ldots,x_n$ for the original equation, that is  $$x_k=\text{cis}(\frac{2k\pi}{n+1}),\ \ k=1,\ldots,n.$$  This is as far as I could go...    Hints and full answers are welcomed. Include chapters in List of Figures with titletoc? There is an option to filter only one color (R,G,B,Y) in my camera. What I want to know is can this effect be achieved in photoshop? And what is this called? I tried googling monochrome and came up with nothing. My camera calls it part color. Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$. Let $f$ be a real-valued function continuous on $[a,b]$ and differentiable on $(a,b)$. Suppose that $\lim_{x\rightarrow a}f'(x)$ exists. Then, prove that $f$ is differentiable at $a$ and $f'(a)=\lim_{x\rightarrow a}f'(x)$.    It seems like an easy example, but a little bit tricky. I'm not sure which theorems should be used in here.    ==============================================================  Using @David Mitra's advice and @Pete L. Clark's notes I tried to solve this proof. I want to know my proof is correct or not.  By MVT, for $h&gt;0$ and $c_h \in (a,a+h)$ $$\frac{f(a+h)-f(a)}{h}=f'(c_h)$$ and $\lim_{h \rightarrow 0^+}c_h=a$.    Then $$\lim_{h \rightarrow 0^+}\frac{f(a+h)-f(a)}{h}=\lim_{h \rightarrow 0^+}f'(c_h)=\lim_{h \rightarrow 0^+}f'(a)$$   But that's enough? I think I should show something more, but don't know what it is. Solve a linear system with more variables than equations What is the different between an Normal user account and the Admin account in Ubuntu?  You must always enter your password if you want do something. Can you give a Normal account  "limitations" to do not install or remove anything or to not open any program? Can you also "hide/deactive" the terminal and console for a normal user? I have the string   a.b.c.d   I want to count the occurrences of '.' in an idiomatic way, preferably a one-liner.  (Previously I had expressed this constraint as "without a loop", in case you're wondering why everyone's trying to answer without using a loop). "Least Astonishment" and the Mutable Default Argument How do I downgrade Firefox v. 57 to v. 56? What is the most straight forward way to downgrade Firefox 57 to Firefox 56?   This is to have more time to wait for working replacements for legacy extensions. Why did they make another prequel? Star Trek: Discovery, the first new Star Trek television series in 12 years, debuted last week.  As with a number of recent Star Trek efforts, this is a prequel to the Original Series, occurring roughly one decade prior chronologically.    The Original Series, the Next Generation, Deep Space Nine, and Voyager occur in a chronological sequence that, generally speaking, moves forward (although TNG, DS9, and Voyager overlap to various degrees).  The linear trajectory of Star Trek productions took a sudden shift with the television series Enterprise, which was set just over 100 years before TOS. This was followed by J.J. Abrams' 2009 film Star Trek, which created a new timeline and which followed the characters from TOS, but at a time that is also chronologically one decade before TOS.  Given that the last few interpretations of Star Trek were prequels in one sense or another, I was disappointed that Discovery was yet another prequel.  I'd like to know why the showrunners decided to take this direction.  I haven't been able to find any statements about the rationale.    That being said, I find it interesting that Star Trek (2009) was set in 2255-2258 and Discovery is currently set in 2256, even though they are ostensibly in different timelines &mdash; the former in the reboot timeline and the latter supposedly in the original timeline.  Also, one of the showrunners for Discovery, Alex Kurtzman, was one of the principal members of the creative team behind the Abrams films.  Why did the creative staff behind Discovery decide to make another prequel?  Does it have anything to do with the fact that Star Trek (2009) and Discovery are set in roughly the same years? How can I get id,class or name attr an element with jquery How to count derangements? I was going through .Suppose that there are n persons who are numbered 1, 2, ..., n. Let there be n hats also numbered 1, 2, ..., n. We have to find the number of ways in which no one gets the hat having same number as his/her number. Let us assume that the first person takes hat i. There are n − 1 ways for the first person to make such a choice. There are now two possibilities, depending on whether or not person i takes hat 1 in return:  A) Person i does not take the hat 1. This case is equivalent to solving the problem with n − 1 persons and n − 1 hats: each of the remaining n − 1 people has precisely 1 forbidden choice from among the remaining n − 1 hats (i's forbidden choice is hat 1).  B) Person i takes the hat 1. Now the problem reduces to n − 2 persons and n − 2 hats.  From this, the following relation is derived: !n = (n - 1) (!(n-1) + !(n-2))  I have a hard time understanding point B and the equation derived. I was trying to understand with an example of 4 people named A, B, C and D and the corresponding hats A, B, C and D. Everyone of them are supposed to wear any hat except their own matching hat i.e. A can wear B, C or D.  If suppose A wears B hat so we will be left with only 3 people and 3 hats. So I can understand "(n-1)!(n-1)" part but what i didn't understand is part B(marked above). Can anyone help? How to add programs to the Unity Launcher or Ubuntu Dock? How can I add new programs to the launcher (or the dock in Ubuntu 17.10 and later) in Ubuntu? Javascript Date: next month I've been using Javascript's Date for a project, but noticed today that my code that previously worked is no longer working correctly. Instead of producing Feb as expected, the code below produces March.   My code looks something like this:   current = new Date(); current.setMonth(current.getMonth()+1); //If today is Jan, expect it to be Feb now   This code worked everyday up until today. Is this a Javascript bug or am I going about this the wrong way? $\sum_{n=-\infty}^{\infty}\frac{1}{(n+\alpha)^2}=\frac{\pi^2}{(\sin \pi \alpha)^2}$? I'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this:  Let $u=x^x$.  $$\begin{align} \int x^x \, dx&amp;=\int u \, du\\[6pt] &amp;=\frac{u^2}{2}\\[6pt] &amp;=\dfrac{\left(x^x\right)^2}{2}\\[6pt] &amp;=\frac{x^{2x}}{2} \end{align}$$  But it's certain that this isn't the correct way to evaluate that, and the answer must be wrong. Is it posible to add iframe directly in lightning components? Is it possible to add iframe directly in a lightning component as below to visible in community sites ?  &lt;aura:component implements="forceCommunity:availableForAllPageTypes" access="global" &gt;     &lt;iframe width="800" height="600" style="border: 1px solid" src="---" /&gt; &lt;/aura:component&gt; A theorem about Cesàro mean, related to Stolz-Cesàro theorem Original Title: Tauberian theorems and Cesàro sum  Theorem (Landau-Hardy, From Rudin's Principle of Mathematical Analysis Exercise 3.14)  $\newcommand\abs[1]{\left\lvert#1\right\rvert}$ If $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n=\frac{s_0+s_1+\dotsb+s_n}{n+1}\qquad(n=0,1,2,\dotsc)$$ Put $a_n=s_n-s_{n-1}$ for $n\ge1$. Assume $M&lt;+\infty$ and $\abs{na_n}\le M$ for all $n$, and $\lim_{n\to\infty}\sigma_n=\sigma$, then $\lim_{n\to\infty}s_n=\sigma$.  The outline of the proof  If $m&lt;n$, then $$s_n-\sigma_n=\frac{m+1}{n-m}(\sigma_n-\sigma_m)+\frac1{n-m}\sum_{k=m+1}^n(s_n-s_k)\tag{*}$$ Notice that $\abs{s_n-s_k}\le(n-m-1)M\,/\,(m+2)$, fix $\epsilon&gt;0$ and associate with each $n$ the integer $m$ that satisfies $$m\le\frac{n-\epsilon}{1+\epsilon}&lt;m+1$$ Then $(m+1)\,/\,(n-m)\le1/\epsilon$ and $\abs{s_n-s_k}&lt;M\epsilon$. Hence $$\limsup_{n\to\infty}\,\abs{s_n-\sigma}\le M\epsilon$$  Questions and thoughts  It seems that the equation (*) comes out strangely. I wonder how to discover such kind of strange identities. So is there any observation, even deeper, to look through that equation?  Thanks!