Can anyone derive the formula for the expansion $(x + \Delta x)^{n}$ that uses Big O notation?
Explanation of the binomial theorem and the associated Big O notation
How to make a file (e.g. a .sh script) executable, so it can be run from a terminal Integration of $\frac {1}{1+\sin x}$ For $\int \frac {1}{1+\sin x} \mathrm{d}x$, my calculus textbook told me to multiply the integrand by $\frac {1- \sin x}{1- \sin x}$ to obtain: $$ \int \frac{1-\sin x}{1-\sin^2 x} \mathrm{d}x= \int \frac{1-\sin x}{\cos^2 x} \mathrm{d}x$$  From here, it is a matter of breaking up the integral to two very trivial integrals, $\int \sec^2x \mathrm{d}x$ and $\int \sec x \tan x \mathrm{d}x$, which are easily found via commonly known rules.  My question is whether there exists a way of finding $\int \frac {1}{1+\sin x} \mathrm{d}x$ without having to perform the multiplication by $1- \sin x$ in the numerator and denominator. How can I add reflection to a C++ application? I'd like to be able to introspect a C++ class for its name, contents (i.e. members and their types) etc. I'm talking native C++ here, not managed C++, which has reflection. I realise C++ supplies some limited information using RTTI. Which additional libraries (or other techniques) could supply this information? Amazon RDS - Online only when needed? I'm just going to throw this out there, but Google indexes Stack Overflow really quickly For example, I  after using Google to try to find an answer. I then turned back to Google and did a search for  and it was already the first result! Similar searches also put my question on the first page, usually in the top 5!  This is insanity. Did I get lucky and post just before an indexing? Is the popularity of SO leading to Google indexing extremely frequently? Or is Jeff using some kind of crazy SEO? In Java, what is the best way to determine the size of an object? I have an application that reads a CSV file with piles of data rows.  I give the user a summary of the number of rows based on types of data, but I want to make sure that I don't read in too many rows of data and cause OutOfMemoryErrors.  Each row translates into an object.  Is there an easy way to find out the size of that object programmatically?  Is there a reference that defines how large primitive types and object references are for a VM?  Right now, I have code that says read up to 32,000 rows, but I'd also like to have code that says read as many rows as possible until I've used 32MB of memory.  Maybe that is a different question, but I'd still like to know. Make a Bash alias that takes a parameter? So this homework question is in the context of $\mathbb{R}$ only, and we are using Lebesgue measure.  The sum $A+B$ is defined to be $A+B=\{a+b|a\in A,b\in B\}$. The question is: If $m(A),m(B)&gt;0$, is it necessarily the case that $A+B$ contains an open interval?  This question is quite annoying, since it is quite difficult to calculate the sum even after you construct an example. However, it seems like no matter how I constructed a set of positive measure, there is a region that have a lot of point in it (yet this region is not necessarily dense). This suggest that when summed up, the holes should be covered up somehow and thus there would be an open interval, and so the answer is yes. Sorry for a rather vague idea, but I'm just not sure what to do with this problem. I have a feeling that somehow topology would have a way to capture precisely the idea above about a region with a lot of point.  (also, I think my middle school teacher mentioned this once, and I remembered the answer to be yes; but that is quite a while ago though)  Thank you. Any help will be appreciated.  EDIT2: So Camilo's answer is gone. I am looking at the convolution method right now. Because the convolution is continuous, if it is nonzero I am done. But then there is no guarantee that the convolution is nonzero even if $x\in A+B$. I still need help with this direction. Can someone give me some hints?  EDIT: Camilo's answer below essentially suggest the method of getting 2 open interval from regularity theorem, one cover a subset of A and one cover a subset of B such that the sum of the measure of these two subset is bigger than the measure of the larger interval. user99680's answer point to a wiki which show a method that essentially simply acquire an open set. After some thinking, I believe that these method simply can't be made to work. I constructed this example in the hope that it would convince you that it is the case:  Let $I_{0},I_{1},\ldots$ be disjoint open interval such that the distance between any 2 of them is nonzero and $m(I_{n})=9^{-n}$ and all of them are inside $[0,2]$. Let $J_{0},J_{1},\ldots$ be disjoint open interval such that the distance between any 2 of them is at least $3$ and $m(J_{3n})=m(J_{3n+1})=m(J_{3n+2})=\frac{m(I_{n})}{3}$. Let $O=\bigcup I_{n}$ and $H=\bigcup J_{n}$. Construct $A\subset O$ such that $m(A\bigcap I_{n})=\frac{3m(I_{n})}{4}$. Construct $B\subset H$ such that $m(B\bigcap J_{n})=\frac{3m(J_{n})}{4}$. Now if you apply Camilo's method, it is possible that your $I$ would be one of the $I_{n}$ above, and your $J$ is one of the $J_{m}$ above. No matter which $I_{n},J_{m}$ you have, the argument does not work. Rewrite $m=3q+r$ where $0\leq r\leq 2$. If $q&lt;n$ then $m(J_{m})&gt;m(I_{n})$ so the longer interval have length $\frac{m(I_{q})}{3}$; $m(A\bigcap I_{n})$ is at most $\frac{3m(I_{q+1})}{4}=\frac{m(I_{q})}{12}$ and $m(B\bigcap J_{m})=\frac{3m(J_{m})}{4}=\frac{m(I_{q})}{4}$. Thus $m(A\bigcap I_{n})+m(B\bigcap J_{m})$ is no larger than the larger interval. If $n\geq q$ then $m(I_{n})&gt;m(J_{m})$ so the larger interval is of length $m(I_{n})$; $m(A\bigcap I_{n})=\frac{3m(I_{n})}{4}$ and $m(B\bigcap J_{m})$ is at most $\frac{3m(J_{3n})}{4}=\frac{m(I_{n})}{4}$. Once again $m(A\bigcap I_{n})+m(B\bigcap J_{m})$ is no larger than the larger interval. Hence the Camilo's argument does not work. user99680's method does not work either, since you get $O$ and $H$ which is even bigger than any interval. What are the strengths and weaknesses of each follower? $f$ differentiable except in $1$ point Assume we have a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that  $f$ is differentiable for all non-zero $x$, $f$ is continuous at $0$ and  $$\lim_{x \uparrow 0} f'(x) = \lim_{x \downarrow 0} f'(x) &lt; \infty. $$  Is it then true that $f$ is also differentiable at $0$?  If we drop the hypothesis that $f$ is continuous, this is not true since the signums function is a simple counterexample. But what if $f$ is continuous? I don't think this is true but I can't find a counterexample... Unable to cover get set in apex test class I am unable to cover the following class get set method. Need help in creating test class  public class Aresult{      public class Parent {         public String type {get; set;}         public String id {get; set;}         public String sequence_id {get; set;}         public String etag {get; set;}         public String name {get; set;}     } }   Please suggest  @isTest  public class CirriusGetItemsResultTest {     public static void myUnitTest(){         Aresult.parent a = new Aresult.parent();         a.type='hello';         a.id='hello';         a.sequence_id ='hello';         a.etag ='hello';         a.name ='hello';     } } Aggregate Rating not displayed in Google search results Lines appear on the texture in render I am trying to learn Blender. While trying to render a subdivided plane with an image texture applied to it, the subdivision edges are not removed in the rendered result.  Kindly, advise, why these edges appear in the render. Proving that if $f: \mathbb{C} \to \mathbb{C} $ is a continuous function with $f^2, f^3$ analytic, then $f$ is also analytic Let $f: \mathbb{C} \to  \mathbb{C}$ be a continuous function such that $f^2$ and $f^3$ are both analytic. Prove that $f$ is also analytic.   Some ideas: At $z_0$ where $f^2$ is not $0$ , then $f^3$ and $f^2$ are analytic so $f = \frac{f^3}{f^2}$ is analytic at $z_0$ but at $z_0$ where $f^2$ is $0$, I'm not able to show that $f$ is analytic. Which is correct, "you and I" or "you and me"? Curved space-time VS change of coordinates in Minkowski space Is it OK to require users to have a third party login account? Translating cooking terms between US / UK / AU / CA / NZ Hypernym for “import” and “export”? Why is the order of $\mathbb{Z}[i]/\langle 3+i\rangle$ equal to 10? The reasoning my book uses is as follows:   Since $(3+i)(3-i) = 10$, $10 + \langle 3+i\rangle = 0 + \langle 3+i\rangle $. Also, $i + \langle 3+i\rangle = -3 + \langle 3+i\rangle = 7 + \langle 3+i\rangle$. So, $\mathbb{Z}[i]/\langle 3+i\rangle = \{k + \langle 3+i\rangle \mid k = 0,1,\ldots ,9\}$, since $1 + \langle3+i\rangle$ has additive order $10$.   I do not understand why the additive order of $1 + \langle 3+i\rangle$ is $10$, or why the fact that $(3+i)(3-i) = 10$, shows the additive order or $1 + \langle 3+i\rangle$  is $10$. If someone could please explain this I would greatly appreciate it.   Note: I apologize for not using MathJax to write this post. I intend to learn it over the coming weekend.