Find $\lim_{n \to \infty}\left(1+\frac{1}{n^2}\right)\left(1+\frac{2}{n^2}\right)\cdots\left(1+\frac{n}{n^2}\right)$
How to calculate $\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$?
HP PSC-2100 can't seem to get xsane and HPLIP (printing working) How can I remove Ubuntu from my system and install the Windows bootloader? UV mapping not filling faces I have tried to map this simple object, but the uv mapping does not correspond to what i see in the textured object. It cuts the image a bit on top and a bit on the bottom of the faces. It does not fill the faces with the texture, even though i have mapped them correctly. Does anyone know what could be happening? Thanks in advance. Punctuation of a Direct Quote Is any particular punctuation necessary when a quotation is the subject of a sentence? For example, in the sentence below, should there be a comma after export? Or anything else?     "Laughter is America's most important export" is a well-known quote of Walt Disney's. Find $(a,b)$ such that $\lim\limits_{x\to 0}\frac{f(x)}{x}=1$ implies $\lim\limits_{x\to 0}\frac{x(1+a\cos x)-b\sin x}{(f(x))^3}=1$ Let $f$ denote any function such that $\lim\limits_{x\to 0}\frac{f(x)}{x}=1$. Find the value of $a$ and $b$, assuming that $\lim\limits_{x\to 0}\frac{x(1+a\cos x)-b\sin x}{(f(x))^3}=1$.   My  attempt: $\lim_{x\to 0}\frac{x(1+a\cos x)-b\sin x}{(f(x))^3}=1$ $\lim_{x\to 0}\frac{x(1+a\cos x)-b\sin x}{x^3\frac{(f(x))^3}{x^3}}=1$ $\lim_{x\to 0}\frac{x(1+a\cos x)-b\sin x}{x^3}=1$ $\lim_{x\to 0}\frac{1+a\cos x-b\frac{\sin x}{x}}{x^2}=1$ $\lim_{x\to 0}\frac{1+a\cos x-b}{x^2}=1$ As denominator is tending to zero,so numerato will also tend to zero. $1+a-b=0............................(1)$ Applying L Hospital rule, $\lim_{x\to 0}\frac{-a\sin x}{2x}=1$ So $a=-2$ and $b=-1$ But the answer given in my book is $a=\frac{-5}{2}$ and $b=-\frac{3}{2}$. I do not understand where have i gone wrong? sequence $x_{n+1}=x_n+e^{-2018x_n}$ How does he have those memories? Can you grab the blue shirts and socks?   Is the above sentence stating that both the shirts and the socks are blue? Or only the shirts?  At this stage, I am leaning towards the earlier (only the shirts) — though writing "Can you grab the blue shirts and blue socks?" seems redundant. How does a circular lens produce rectangular shots? If a topology contains all infinite subsets, then it is the discrete topology I am solving question $8$ of exercises in section 1.1. of chapter 1 in "Topology without tears". The question reads as follows.     Let $X$ be an infinite set and $\tau$ be a topology on $X$. If every infinite subset of $X$ is in $\tau$, prove that $\tau$ is the discrete topology   My idea was to show that every singleton set is in the topology and thereby the topology is discrete.  Choose an infinite set $A$ from $X$ such that $A^c$ is also infinite. $A$ and $A^c$ belong to $\tau$ since they are infinite sets. Now consider any $x \in X$. Note that $A \cup \{x\}$ and $A^c \cup \{x\}$ belong to $\tau$ since they are infinite sets. Hence, their intersection which is nothing but $\{x\} \in \tau$ for every $x \in X$.  The problem I have is I don't know how to prove the first sentence in the previous paragraph. These are my line of thoughts to prove them.   If $X$ is a countably infinite set, then I can list the elements are let the odd numbered elements fall into $A$. This guarantees $A$ and $A^c$ are both infinite. If $X$ is uncountable, then I can choose a countably infinite subset and call it $A$.   Are the above arguments rigorous? I am not especially happy with my second argument of choosing a countably infinite subset from an uncountable set since I do not give an explicit procedure of constructing the set $A$.  Also, is there any other simpler way of answering the original question? How to prove that $\hat\sigma^2$ has $\chi^2_{n-p}$ distribution (linear regression) Consider the linear regression model:   $$Y_i=r(x_i)+\varepsilon_i\equiv\sum_{j = 1}^p x_{ij} \beta _j + \varepsilon _i,\quad i=1,\ldots,n.$$  where $x_1,\ldots,x_n\in \mathbb{R}^p$ are fixed, $E(\varepsilon_i)=0$,  $\operatorname{Var}(\varepsilon_i)=\sigma^2$. Denote $Y=(Y_1,\ldots,Y_n)^T$, $\beta=(\beta_1,\ldots,\beta_p)^T$, $X=(x_{ij})_{n\times p}$. As is known, $\hat \beta  = \arg\min\limits_{\beta  \in \mathbb{R}^p} (Y - X\beta)^T (Y - X\beta)=(X^TX)^{-1}X^TY$ if the matrix $X^TX$ is invertible, and so an estimator of $r(x)$ at $x=(x_1,\ldots,x_p)\in\mathbb{R}^p$ is given by $\hat r_n(x)=x^T\hat\beta$.  Let $$\hat{\sigma}^2 = \frac{1}{\sigma^2}\sum_{i = 1}^n (Y_i - {\hat r}_n (x_i))^2.$$ I am stucking the problem: $\hat\sigma^2$ has $\chi^2_{n-p}$ distribution. How to prove the statement? $ \int_{-2}^{0} \frac{x}{\sqrt{e^x+(x+2)^2}} dx $ = -$  2.887270... $  The function has no antiderivate and there's no symmetry here to help, if you solve this I would be thankful if you would for a highschooler ^^ What steps can I take to try to recover lost or inaccessible data from any storage device?  Answers:   This applies to any computer storage device, e.g. internal/external hard drives, USB sticks, flash memory. The most important thing is to STOP using it, any type of I/O can ruin your chances of a recovery.     We have separate questions covering common problems with USB flash drives in greater detail: Geometric series and big theta Consider the following function:  $$S(n)=1+ c + c^2 + ··· + c^n,$$  where c is a positive real number.  (A) This function is the sum of a geometric series. Give a precise closed-form formula for $S(n)$, interms of $c$ and $n$, in the case where $c \neq 1$.  (B) Show that $S(n)$  is:  $\theta(1)$ if $c&lt;1$  $ \theta(n)$ if $c=1$  $\theta(c^n)$ if $c&gt;1$    can someone please explain how i can go about finding the solutions for these problems, i have never encountered these and am stuck, i know that big theta means that the function is bond on both sides but how do i got about finding the answer?  (edit) im still having trouble understanding part B. can someone please explain? (can someone walk me through the second one) so far i have $n \leq c^{n+1} \leq ???$  (edit #2) i tried doing the first one can someone verify if i did it right  $S(n)$ is $\Theta(1)$ if c&lt;1  (1)$S(n)$ is $O(1)$ if c&lt;1  (2)$S(n)$ is $\Omega(1)$ if c&lt;1  (1) $$\frac{C^{n+1}-1}{C-1} \lt (C_1)(1)$$ $$\frac{C^{n+1}-1}{C-1} \lt (C_1)$$ $$C^{n+1} \lt C_1(C-1)$$ $$C^nC-1 \lt C_2C-1$$ $$C^n \lt C_2$$ $$C^n \lt 1$$ since c&lt;1 this means that any number that will be $C^n$ will be less then 1 for all positive n  i did the omega part pretty much the same way and got $$C^n \gt C_2$$ $$C^n \gt 0$$ since c&lt;1, $C^n$ can potentially become very very small so the only thing bounding it is 0 for all positive n Is a contradiction enough to prove a set equality to $\varnothing$? Why is radian so common in maths? I have learned about the correspondence of radians and degrees so 360° degrees equals $2\pi$ radians. Now we mostly use radians (integrals and so on)  My question: Is it just mathematical convention that radians are much more used in higher maths than degrees or do radians have some intrinsic advantage over degrees?  For me personally it doesn't matter if I write $\cos(360°)$ or $\cos(2\pi)$. Both equals 1, so why bother with two conventions? Unhandled event loop exception in plugin org.eclipse.ui Allowing users to compile / run PLSQL and SQL As a DBA - assuming my users are knowledgeable with PLSQL and SQL - why would I not grant my users privileges to compile and run their own PLSQL code. We have an MI database environment where the data has undergone ETL. The tables contain a large numbers of rows but not necessarily large rows in size. The analysis - will be - I assume of calculus in nature. The users are wanting to run their own code on the data for the purposes of ad-hoc analysis - within their own schemas. They come from a SAS background which from my understanding has allowed them to write and run their own analysis of data. Albeit, SAS runs on its own dataset - extracted from Oracle or elsewhere.  In my opinion this is a dangerous set-up - but without answering my own question - what are the pitfalls? Installing older iOS for device testing Installing titling.sty onto an offline Windows machine I'm trying to compile a file for which I need titling.sty:  ! LaTeX Error: File `titling.sty` not found.  The two biggest issues are 1) it's on a Windows machine (I'm really only &quot;comfortable&quot; with the Unix command line) and 2) the machine cannot be connected to the internet (so automatic approaches are impossible). Assuming the error means I'm missing a package and following some related advice (e.g., ), I downloaded the file , transferred it to the offline machine, and unzipped it, then tried to run: latex titling.ins  But this gave me an error (given I can't copy-paste, I'm including what I think may be relevant):  ******************************************************** * No Configuration file found, using default settings. * ******************************************************** ! ************************************************** * Could not find your \batchfile=titling.ins. **************************************************. \processbatchFile ...****************************}                                                  \fi \else \ifDefault \Msg ... 1.1092 \process@first@batchfile  I have no idea how to proceed from this. How can I install titling.sty on this machine?