How to put right brace to group cases without using math mode?
How to put a brace on the right, not left, to group cases?
How do you change from icon to list view in 20.4 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 Compute: $\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)$ where $A_n = \frac{1}{n}(a_1 + a_2 + \cdots + a_n)$ Let $\{a_n\}, {n\geq 1}$, be a sequence of real numbers satisfying $|a_n|\leq 1$ for all $n$. Define  $$A_n = \frac{1}{n}(a_1 + a_2 + \cdots + a_n),$$ for $n\geq 1$. Then find $\displaystyle\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)$ .    I proceed in this way  $$\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)=\lim_{n \rightarrow \infty}\sqrt{n}\left[\frac{1}{n+1}(a_1 + a_2 + \cdots + a_n+a_{n+1})-\frac{1}{n}(a_1 + a_2 + \cdots + a_{n})\right]=\lim_{n \rightarrow \infty}\left[{(na_{n+1}-a_1 - a_2 - \cdots - a_n})\frac{1}{\sqrt{n}(n+1)}\right]$$  Please help me to complete from here A generic list of anonymous class Suppose the terms of the sequence of real numbers $\{a_n\}$ satisfy $|a_{n+1} - a_n| \le 2^{-n}$ for all $n$. Prove that $\{a_n\}$ is Cauchy.   My Work  So by the definition of a Cauchy sequence, for all $\varepsilon &gt; 0$ $\exists N$ so that for $n,m \ge N$ we have $|a_m - a_n| \le \varepsilon$. However, questions like  make me understand that the $2^{-n}$ condition is necessary for this to be a true statement.   So I am wondering how to appeal to the Cauchy definition for this proof. Do I prove that every convergent sequence is therefore Cauchy, and then try to prove convergence? How did Montague Apparate into the Hogwarts bathroom? In year 5, Fred and George jam Montague into the Vanishing Cabinet. He's found weeks later in a toilet.      ‘He never managed to get all the words out,’ said Fred, ‘due to   the fact that we forced him head-first into that Vanishing Cabinet   on the first floor.’      […]      ‘They’ve found Montague, sir, he’s turned up jammed inside a toilet   on the fourth floor.’   ‘How did he get in there?’ demanded Snape.   ‘I don’t know, sir, he’s a bit confused.’   In year 6 Malfoy expands on that.      ‘I had to mend that broken Vanishing Cabinet that no one’s used for years. The one Montague got lost in last year.’      […]      ‘The other’s in Borgin and Burkes,’ said Malfoy, ‘and they make a kind of passage between them. Montague told me that when he was stuck in the Hogwarts one, he was trapped in limbo but sometimes he could hear what was going on at school, and sometimes what was going on in the shop, as if the Cabinet was travelling between them, but he couldn't make anyone hear him ... in the end he managed to Apparate out, even though he’d never passed his test. He nearly died doing it.   What happened here? Did he Apparate out into Hogwarts, Apparate into Borgin and Burkes and somehow wander back to Hogwarts and get stuck into a toilet? How would you explain covariance to someone who understands only the mean? ...assuming that I'm able to augment their knowledge about variance in an intuitive fashion (  ) or by saying: It's the average distance of the data values from the 'mean' - and since variance is in square units, we take the square root to keep the units same and that is called standard deviation.  Let's assume this much is articulated and (hopefully) understood by the 'receiver'. Now what is covariance and how would one explain it in simple English without the use of any mathematical terms/formulae? (I.e., intuitive explanation. ;)  Please note: I do know the formulae and the math behind the concept. I want to be able to 'explain' the same in an easy to understand fashion, without including the math; i.e., what does 'covariance' even mean? What are some important applications of the Erdős-Szekeres Theorem? I'm new to regular expressions and would appreciate your help.  I'm trying to put together an expression that will split the example string using all spaces that are not surrounded by single or double quotes. My last attempt looks like this: (?!") and isn't quite working. It's splitting on the space before the quote.  Example input:   This is a string that "will be" highlighted when your 'regular expression' matches something.   Desired output:  This is a string that will be highlighted when your regular expression matches something.   Note that "will be" and 'regular expression' retain the space between the words. Why can't I connect to Pokemon Go via mobile data? Unmarshal a single element list fails What does rate of change actually mean? If we have a function, say, $y=x^2$, then $dy/dx=2x$. Now $y=1,4,9,16...$ for $x=1,2,3,4...$ and $dy/dx=2,4,6,8...$ for $x=1,2,3,4...$ Now as $dy/dx$ represents rate of change of y w.r.t x but I can't understand how these particluar values of $dy/dx$ represent change ? I mean for x=3, the rate of change is 6. What does it mean ? Floating point arithmetic not producing exact results Homomorphisms from $\mathbb{Z} \rightarrow S_3$. Is it a scam if the person only wants to deposit into my account, not make a withdrawal? A man in  west  coast  Austria wants to deposit money  into  my account.  I'm in Alabama.  I had to get an online  account.  I did that, but he says his bank manager  needs all this information:   DOB SSN    online access username  online access  password   security  question and answer   bank name   bank address   bank number  person who is on the account     their name their address    account no   routing no     OK. Is it safe  to do this or am I being scammed?  I don't want him  to be able to end up withdrawing all my money instead of depositing money. How do I share an ethernet connection via wifi on Windows 8? I have a Windows 8 laptop that is connected to a router via Ethernet. I want to make the laptop act like a wireless router via wifi so that other devices could connect to the computer using wifi and gain access to its Ethernet internet connection.  How do I do something like this? I have a vector file containing thousands of points. Where points are within x distance from each other, I wish to delete one of the points and keep the other (either by snapping to one point, or simply erasing one of the two points)  However, in Whitebox the vector cleaning tool only works with polygons and lines.  Is there some other way I can do this with my shapefiles? Project Euler Question 14 (Collatz Problem) Help with this trigonometry problem Prove the given identity.  $$\left(\sin\frac {9\pi}{70}+ \sin\frac {29\pi}{70} - \sin\frac {31\pi}{70}\right) \left(\sin\frac {\pi}{70}-\sin\frac {11\pi}{70} - \sin\frac {19\pi}{70}\right) =\frac {\sqrt {5} -4}{4}$$  Please help, I could not gather enough ideas, even on how to get to the first step. I thought of using the transformation formula (from sum to product) but did not find a fruitful result.   UPDATE 1.:  $\left(-\frac{1 +\sqrt{5}}{8} + \frac{1}{8} \sqrt{14\left(5 - \sqrt{5} \right)}\right)\left(-\frac{ 1+ \sqrt{5}}{8} -\frac{1}{8} \sqrt{14 \left(5 -\sqrt{5} \right)}\right) $,    I noticed that Alexis had got to this point.  Can anyone explain me how did he get here, with calculations. Please help.  Thanks in Advance. How can I combine multiple rows into a comma-delimited list in Oracle? I have a simple query:  select * from countries   with the following results:  country_name ------------ Albania Andorra Antigua .....   I would like to return the results in one row, so like this:  Albania, Andorra, Antigua, ...   Of course, I can write a PL/SQL function to do the job (I already did in Oracle 10g), but is there a nicer, preferably non-Oracle-specific solution (or may be a built-in function) for this task?  I would generally use it to avoid multiple rows in a sub-query, so if a person has more then one citizenship, I do not want her/him to be a duplicate in the list.  My question is based on the similar question on .  UPDATE: My function looks like this:  CREATE OR REPLACE FUNCTION APPEND_FIELD (sqlstr in varchar2, sep in varchar2 ) return varchar2 is ret varchar2(4000) := ''; TYPE cur_typ IS REF CURSOR; rec cur_typ; field varchar2(4000); begin      OPEN rec FOR sqlstr;      LOOP          FETCH rec INTO field;          EXIT WHEN rec%NOTFOUND;          ret := ret || field || sep;      END LOOP;      if length(ret) = 0 then           RETURN '';      else           RETURN substr(ret,1,length(ret)-length(sep));      end if; end;