Prove that   $$ \lim_{n \to \infty}\int_0^{n^2} n\sin(x/n)e^{-x^2} dx = 1/2.$$   I've tried all of the techniques learned in the course so far and none of them seem to work, so if anyone could help me with this it would be much appreciated. Thanks!
I am trying to compute this integral/limit, I don't feel like I have any good insight...   $$\lim_{n \to \infty} \int_0^{n^2} e^{-x^2}n\sin\frac{x}{n} \, dx.$$  I have tried to make a change of variable to get rid of the $n^2$, I changed to $X=\frac{x}{n^2}$ but got something even worse, I've tried to reach a situation where I could use a convergence theorem for Lebesgue Integrals,... I'm not sure I'm even on the right track!  Could you give me a hint on how to start this?  Thank you very much!
Is there a general formula for image size vs. print size? What size images do I need for common print sizes, such as wallet size, 8x10, and 16x20?  Is there a general formula?  How could that size change if, for example, a 16x20 was printed on canvas rather than photopaper? Are IP addresses with and without leading zeroes the same? Transforms moving around the 3D cursor? I think I accidentally pressed something to make it like this:    See how the XYZ arrows are at the 3D cursor instead of those vertexes I've selected?  How do I fix that? Whenever I select on a vertex or something, those blue, red, green arrows would always be where I selected, now it just keeps going to the 3D cursor. In MTG, what happens when a player loses in multiplayer? Mouse rotated 90 degrees Why the rows returns by "explain" is not equal to count()? What are "schlieren" in English? The German word for these patterns is "Schliere, pl. Schlieren", and apparently this word has entered the English language as a loanword (cf. article  in Wikipedia or entry  in OED). But the German loanword is probably not something most native speakers of English understand without the help of a dictionary, so what would you call, or how would you describe, the following patterns?  Concrete:    Air:    Oil on water:    Coffee and milk:    Paint:    Smoke:    Rock:    Marbled paper:    And so on.    "Waves" is not applicable, because waves are vertical undulations of the surface, while these are a pattern on, or inside, the surface level, with no amplitude.  "Striæ" does not really fit either, because those refer more to the parallel layers, than to the curly forms. What would be the differences between a persistent USB Live Session and a installed Ubuntu in a USB drive? I'm a teacher and will ask my students to either bring their laptops with Ubuntu installed with an specific list of packages preinstalled or bring a USB Drive or External Hard Drive with Ubuntu and the packages preinstalled.  For those who are going to bring the USB are two posibilities:   Install with the Ubuntu's "Startup disk creator" program, and telling it to reserve some space for persistence. Boot with a LiveCD and Install Ubuntu in the USB drive.   Then, in either case, boot with that media and install the packages.  So, for my purpose or any other purpose:   What is the best choice and why? Do we have another option? full explanation please. 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? Search window not opening in ArcMap? Is there an easy way to see associativity or non-associativity from an operation's table? Most properties of a single binary operation can be easily read of from the operation's table. For example, given $$\begin{array}{c|ccccc} \cdot &amp; a &amp; b &amp; c &amp; d &amp; e\\\hline a     &amp; e &amp; d &amp; b &amp; a &amp; c\\ b     &amp; d &amp; c &amp; e &amp; b &amp; a\\ c     &amp; b &amp; e &amp; a &amp; c &amp; d\\ d     &amp; a &amp; b &amp; c &amp; d &amp; e\\ e     &amp; c &amp; a &amp; d &amp; e &amp; b \end{array}$$ it is easy to check that it is closed (no elements occur in the table which don't occur as row or column index), commutative (the table is symmetric), has a neutral element (the row and column of $d$ are copies of the index row/column), and has an inverse element for each element (there's a $d$ in each row and column). In other words, almost all important properties can immediately be seen. The only part missing is associativity.  Therefore my question: Is there a simple way to see directly from the operation's table (i.e. without doing explicitly all the calculations) if an operation is associative? Tachyons and Photons Is there a particle called a "tachyon" that can travel faster than light?  If so, would Einstein's relativity be wrong?  According to Einstein no particle can travel faster than light. How do I find the privileges I have earned and will earn in Stack Overflow? It used to be off the top bar, now its gone  Where has it moved?  EDIT:  It moved onto the profile page !  Right side same row as your name in profile:    Thank you Prove by induction that for all $n \ge 0$:  $${n \choose 0} + {n \choose 1} + ... + {n \choose n} = 2^n.$$  In the inductive step, use Pascal’s identity, which is:  $${n+1 \choose k} = {n \choose k-1} + {n \choose k}.$$  I can only prove it using the binomial theorem, not induction. Showing that the orthogonal projection in a Hilbert space is compact iff the subspace is finite dimensional Suppose that we have a Hilbert Space $H$ and $M$ is a closed subspace of $H$. Let $T\colon H\rightarrow M$ be the orthogonal projection onto $M$.   I have to show that $T$ is compact iff $M$ is finite dimensional.  So if we assume that $M$ is finite dimensional then $\overline{T(B(0,1))}$ is a closed bounded set in a finite dim vector normed space and so it is compact. Which gives that $T$ is compact.  But I am unsure how to prove that if $T$ is compact then $M$ is finite dimensional?  Thanks for any help Benchmarking hosting providers IO with Bonnie What is the method for converting radians to degrees? I run into this occasionally and always forget how to do it.  One of those things that pop up ever so often.  Also, what's the formula to convert angles expressed in radians to degrees and back again? When we say something in specified operating system, which preposition should I use?  e.g.     I play games {in/on/under} Windows. Is there a way to conveniently define a C-like structure in Python? I'm tired of writing stuff like:  class MyStruct():     def __init__(self, field1, field2, field3):         self.field1 = field1         self.field2 = field2         self.field3 = field3 Expectation and variance of the geometric distribution How can one use memorylessness and the  and the  to find the expectation and variance of the geometric distribution?  I will post my own answer, but as always, that shouldn't stop anyone else from posting theirs.