Synthetic division for: $\frac{60 x^{3}+43x^{2}-34x-24}{3x+2}$
Synthetic division of polynomials by factor of the form $(ax+b)$
I have been thinking on and off about a problem for some time now. It is inspired by an exam problem which I solved but I wanted to find an alternative solution. The object was to prove that some sequence of functions converges weakly to zero in $L^2$.   I managed to show (with some help) that the limit $f$ (of a subsequence) satisfies $\int_0^x f \ dm=0$ for all $x&gt;0 $. From this I want to conclude that $f=0$ a.e. I can do this with the fundamental theorem of calculus in its Lebesgue version but there ought to be a more elementary proof.  Can someone here help me out? Convergence of the sequence $\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots$ I recently came across this problem     Q1 Show that $\lim\limits_{n \rightarrow \infty} \underbrace{{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}}}_{n \textrm{ times  }} = 3$   After trying it I looked at the solution from that book which was very ingenious but it was incomplete because it assumed that the limit already exists.  So my question is     Q2 Prove that the sequence$$\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots,\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}$$ converges.   Though I only need solution for Q2, if you happen to know any complete solution for Q1 it would be a great help .  If the solution from that book is required I can post it but it is not complete as I mentioned.   Edit: I see that a similar question was asked before on this site but it was not proved that limit should exist. Skipping Iterations in Python I have a loop going, but there is the possibility for exceptions to be raised inside the loop. This of course would stop my program all together. To prevent that I catch the exceptions and handle them. But then the rest of the iteration runs even though an exception occurred. Is there a keyword to use in my except: clause to just skip the rest of the current iteration? How can I get the HTML source of a given web address in C#? Is there are rule when to use opened vs open?  I always get confused even though I've been speaking English as the dominant language for more than half my life.  E.g.         Is the door open(ed)?   Which file do you have open(ed) in your editor? My laptop's cable broke and sadly it seems like I need to buy a new power supply.  What I wonder is, I've found that universal power supplies are much cheaper than the official HP ones. Are they really worth it?  Another question: the original supply has these specs: 65W, 18.5V, 3.5A. However the universal supplies are more powerful (90W, 100W) and come in integer voltages (17V, 18V, 19V, etc). Are they compatible? Should I set them to 18V or 19V? Consistency of an order statistic in exponential distribution Function to determine if two numbers are nearly equal when rounded to n significant decimal digits I have been asked to test a library provided by a 3rd party. The library is known to be accurate to n significant figures. Any less-significant errors can safely be ignored. I want to write a function to help me compare the results:  def nearlyequal( a, b, sigfig=5 ):   The purpose of this function is to determine if two floating-point numbers (a and b) are approximately equal. The function will return True if a==b (exact match) or if a and b have the same value when rounded to sigfig significant-figures when written in decimal.   Can anybody suggest a good implementation? I've written a mini unit-test. Unless you can see a bug in my tests then a good implementation should pass the following:  assert nearlyequal(1, 1, 5)  assert nearlyequal(1.0, 1.0, 5)  assert nearlyequal(1.0, 1.0, 5)  assert nearlyequal(-1e-9, 1e-9, 5)  assert nearlyequal(1e9, 1e9 + 1 , 5)  assert not nearlyequal( 1e4, 1e4 + 1, 5)  assert nearlyequal( 0.0, 1e-15, 5 )  assert not nearlyequal( 0.0, 1e-4, 6 )    Additional notes:   Values a and b might be of type int, float or numpy.float64. Values a and b will always be of the same type. It's vital that conversion does not introduce additional error into the function. Lets keep this numerical, so functions that convert to strings or use non-mathematical tricks are not ideal. This program will be audited by somebody who is a mathematician who will want to be able to prove that the function does what it is supposed to do. Speed... I've got to compare a lot of numbers so the faster the better. I've got numpy, scipy and the standard-library. Anything else will be hard for me to get, especially for such a small part of the project. Recently, I read this  about how they have some of their infrastructure laid out. I was curious to know, does Stack Exchange use caching (both at the application layer and/or database layer) and if so, how? What is $P(X > 0 \mid X + Y > 0)$ given that $X, Y$ are i.i.d standard normal? $G$ solvable $\implies$ composition factors of $G$ are of prime order. I'm trying to prove the equivalency of the following definitions for a finite group, $G$:  (i) $G$ is solvable, i.e. there exists a chain of subgroups $1 = G_0  \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_s = G $ such that $G_{i+1}/ \ G_i$ is abelian for all $0 \leq i \leq s-1$  (ii) $G$ has a chain of subgroups $1 = H_0  \trianglelefteq H_1 \trianglelefteq \cdots \trianglelefteq H_s = G $ such that $H_{i+1} /\  H_i$ is cyclic for all $0 \leq i \leq s-1$.  (iii) All composition factors of $G$ are of prime order.  (iv) G has a chain of subgroups $1 = N_0  \trianglelefteq N_1 \trianglelefteq \cdots \trianglelefteq N_t = G $ such that each $N_i$ is a normal subgroup of $G$, and $N_{i+1}/ \ N_i$ is abelian for all $0 \leq i \leq t-1$.  It was simple to show that (iii) $\implies$ (ii) and (ii) $\implies$ (i). Now, I'm trying to prove that (i) $\implies$ (iii), and I'm getting stuck.   I know that If $G$ is an abelian simple group, then $|G|$ is prime, so all I need to show is that each $G_{i+1}/ \ G_i$ in the definition of solvable is simple.  If I knew that $1 = G_0  \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_s = G $ were a composition series, then each $G_{i+1}/ \ G_i$ would be simple by definition. But the definition of solvable doesn't assume a composition series. I know every group has a unique composition series, but I'm not sure how to connect that to the subgroups in the definition of solvable.   More generally, I guess I just don't understand how being abelian connects to being of simple or of prime order.  Any pushes in the right direction would be appreciated! Show reference label when it is undefined When a reference is undefined, LaTeX shows simply [??] which doesn't give much information. I would like it to show the label instead. This is useful when more collaborators work on the same preprint and the references are yet to be done.  I'm using neither biblatex nor natbib. I do use bibtex but it is irrelevant for this question.  MWE:  \documentclass{article}  \begin{document}  This: \cite{undef} should look like: [{\bfseries ?? undef}]  \end{document} Because of  I'd like to prevent apt-get upgrade and Update Manager from updating the "libgtk2.0-0" package.   How can this be achieved? Generating a range of numbers in MySQL How do I generate a range of consecutive numbers (one per line) from a MySQL query so that I can insert them into a table?  For example:  nr 1 2 3 4 5   I would like to use only MySQL for this (not PHP or other languages). Somewhere in the provided answer:   $$\int \frac{1}{\sqrt{2-x^2}} dx = \sin^{-1}{\frac{x}{\sqrt{2}}}$$  How did they get that? What I have:   $$\frac{1}{\sqrt{2-x^2}} = \frac{1}{\sqrt{2(1-\frac{x^2}{2})}} = \frac{1}{\sqrt{2} \sqrt{1-\frac{x^2}{2}}}$$  $$\frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{1-(\frac{x}{\sqrt{2}})^2}} = \frac{1}{\sqrt{2}} \sin^{-1}{\frac{x}{\sqrt{2}}}$$  So I have an extra $\frac{1}{\sqrt{2}}$ ... I probably had some stupid mistakes? "exec: gnome-session: not found" when logging in using lightdm On login the password is accepted but I'm returned to the login screen. Tried multiple times even after reboot... same behavior.  I left my laptop (Lenovo Ideapad z510) in suspend mode (lid down) last night, but it did not resume this morning, as it usually does. It had shutdown or crashed, and on booting with Ubuntu the login failure mentioned above started.  cat ~/.xsession-errors /usr/sbin/lightdm-session: 1: /usr/sbin/lightdm-session: ls: not found /usr/sbin/lightdm-session: 104: exec: gnome-session: not found   Okay. I've tried a bunch of suggestions from    Changed id to /usr/sbin/id in /etc/profile removed .Xauthority removed /tmp removed,  purged and re-installed lightdm checked .profile for syntax errors (source .profile worked fine, so I guess no syntax errors)   Nothing worked. My guest login is working fine, so another user should also work. I'm now going to create another user and continue working, but a solution to this problem is still warranted... it's likely going to repeat in future.  Installing and reconfiguring gdm also did not work.  sudo apt-get install gdm If $a,b,c$ are positive , show that $$\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}+\dfrac{a^2+b^2}{a+b} \ge a+b+c$$  Trial: Here  I proceed in this way $$\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}+\dfrac{a^2+b^2}{a+b} \ge  \dfrac{2bc}{b+c}+\dfrac{2ca}{c+a}+\dfrac{2ab}{a+b}$$ then how I proceed. Please help. Is convex surface normal in cast iron skillet? I am new to cast iron skillet. I have cooked with it several times and found the Lodge 12" cast iron skillet has a convex surface which is not observable unless you pour oil on it.  When I cook, the oil move to rim of the skillet and makes the center short of oil. Is it normal?  EDIT: I added the photo of the stove and skillet for reference Convert NSDate to NSString How to understand intuitively the Stolz-Cesaro Theorem for sequences?