Define   $x_n=\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\left(1-\frac{1}{10}\right)^2\left(1-\frac{1}{15}\right)^2...\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$ for $n&gt;1$   then $\lim_{n\to\infty} x_n $ is  A)..$\frac{1}{3}$  B)..$\frac{1}{9}$  C)..$\frac{1}{81}$  D)..$0$  What I did is...  $X_n$=$(\prod_{i=2}^{i} \frac{i^2+i-2}{i(i+1)})^2$=$(\prod_{i=2}^{i} \frac{(i+2)(i-1)}{i(i+1)})^2$=$(\frac{4.5.6.....(i+2)()1.2.3.4....(i-1)}{(1.2.3.4...i)(2.3.4.5...(i+1))})^2=(\frac{1}{3}(1+\frac{2}{i}))^2$  as $i\to\infty$ we get $\lim_{x\to\infty} x_n=\frac{1}{9}$  Am I correct or wrong? Any help would be appreciated.
How can I find the following limit :$$\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\left(1-\frac{1}{10}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$$There are four options $(a)\frac1{3}(b)\frac1{9}(c)\frac1{81}(d)0$ Thanks!
Vertical alignment of align* in enumerate Related (but not the same):         Earlier today I asked a question about aligning equations in an enumerate environment and got an answer that seemed to work for the example I gave, but it turns out I wasn't really specific enough for my problem.  I'm using align* environments, often as the only part of a list item. (If you're wondering why I'm not using more words to introduce the align*, these are mostly homework solutions where most of the work is algebra, etc.)  Here's my code and what it makes (above the middle line):  \begin{enumerate}   \item \begin{align*}     x^2 + y^2 &amp;= x^2 + (iy)^2 \\               &amp;= (x + iy) (x - iy)   \end{align*}   \item \begin{align*}     \frac{1}{n^2-4} &amp;= \frac14 \frac{4}{(n-2)(n+2)} \\                     &amp;= \frac14 \left( \frac{1}{n-2}-\frac{1}{n+2} \right)   \end{align*} \end{enumerate}     I'm trying to make the baseline of the first line of the align* environment the same as the number's baseline—I want the first line of the align* to have the same vertical position (centered horizontally in the same way) relative to its number as in the second half of my picture. (Side question: why is there so much space before the align* starts?) My LAN server won't let my cousin join me or me join my cousin. It won't show up on the multiplayer screen - it keeps saying "scanning for LAN servers" but it never finds it. I am at his house and we use different computers but it won't connect. How can I fix this? Showing that $\Bbb R[x] / \langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$ question How do I create a script file for terminal commands? Writing XML to a file without overwriting previous data I tried to install OracleJDK 8 using   sudo add-apt-repository ppa:webupd8team/java   But it's not working.   I searched the problem and found that      "The Oracle JDK License has changed for releases starting April 16, 2019." And  "Oracle Java (JDK) 8 Installer PPA (DISCONTINUED)"   So I installed the OpenJDK8 .   But why oracle cancelled the license of oracle 8 installer? I am on a Mac with MacTex 2020 and ran into a strange problem with Biber. Working on a document all was fine, I did not change the bibliography (says my git) and only fixing typos (not even adding cites) and suddenly latexmk stopped working after a Biber run, while that looked like  INFO - This is Biber 2.14 INFO - Logfile is 'doc.blg' INFO - Reading 'doc.bcf' INFO - Found 31 citekeys in bib section 0 INFO - Processing section 0 INFO - Globbing data source 'literature.bib' INFO - Globbed data source 'literature.bib' to literature.bib INFO - Looking for bibtex format file 'literature.bib' for section 0 INFO - LaTeX decoding ... INFO - Found BibTeX data source 'literature.bib'   so harmless in my view. Then with --debug I additionally get in the end.  INFO - LaTeX decoding ... warning: overriding existing definition of macro "jun" warning: overriding existing definition of macro "jul" warning: overriding existing definition of macro "sep" warning: overriding existing definition of macro "jan" [...] warning: overriding existing definition of macro "nov" INFO - Found BibTeX data source 'literature.bib' Unicode::UCD: failed to find unicore/version in /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc/lib /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc CODE(0x7fd026aeb5a8) CODE(0x7fd026aebb90) at /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc/lib/Biber/Input/file/bibtex.pm line 1011.   It seems unrelated to the library itself since deleting a few entries (going through deleting one half, then the other), the problem persists. It does not happen for an empty literature, but as soon as a first entry is added (just author/journal/title/year/volume/pages - no utf8-only chars) this error message appears.  what does this last error message mean?  How can I solve this? Short story as a letter to original colonisers of earth Show: $ f(a) = a,\ f(b) = b \implies \int_a^b \left[ f(x) + f^{-1}(x) \right] \, \mathrm{d}x = b^2 - a^2 $ How to make IntelliJ IDEA recognise code created by macros? What was the joke that Geordi told Data in TNG: "Encounter at Farpoint"? states:     Apparently at some point during this mission, Geordi La Forge told Data a joke, which the humorless android did not "get" until years later, when he activated his emotion chip during the events of Star Trek: Generations.    Is it known what the joke is? Which way is up? (electric outlet) pgfplots - draw boxplot without outliers I am creating a boxplot with pgfplots, using the builtin statistics library. I load the input data from a table file.  \usepackage{pgfplots} \usepgfplotslibrary{statistics} ... \begin{tikzpicture} \begin{axis}[boxplot/draw direction=y]    \addplot+[boxplot] table[      col sep=tab,      x expr=\coordindex,      y=aColName] {aFileName}; \end{axis} \end{tikzpicture}   Is it possible to draw the plot without outliers? Or hide them? I don't want to pre-calculate the boxplot values by hand. The final state of 1000 light bulbs switched on/off by 1000 people passing by Why Doesn't C# Allow Static Methods to Implement an Interface? Why was C# designed this way?  As I understand it, an interface only describes behaviour, and serves the purpose of describing a contractual obligation for classes implementing the interface that certain behaviour is implemented.  If classes wish to implement that behavour in a shared method, why shouldn't they?  Here is an example of what I have in mind:  // These items will be displayed in a list on the screen. public interface IListItem {   string ScreenName();   ... }  public class Animal: IListItem {     // All animals will be called "Animal".     public static string ScreenName() {         return "Animal";     } .... }  public class Person: IListItem {      private string name;      // All persons will be called by their individual names.     public string ScreenName() {         return name;     }      ....   } Why does my item name not show up in 1.13? How to typeset subscript in usual text mode? How to animate noise texture "seed"? It's possible to animate a change noise texture by animating the texture coordinates with a mapping node or by animating the texture space, however this only works on one axis at a time (on the other axes the texture will appear to be sliding across the object's surface).  For example, the top face has the desired effect but the other faces appear to be moving/sliding upwards:    Even with all three axes location and rotation animated it still appears to be sliding rather than statically changing.    Is there some way to make each face appear like the top face in the first gif but still be seamlessly blended with the other faces? This should work for other shapes as well.  In other words, I'm looking for a way to animate the "seed" of the texture. Let $x_1,x_2,\ldots,x_n$ be the roots for $1+x+x^2+\ldots+x^n=0$. Find the value of   $$P(1)=\frac{1}{x_1-1}+\frac{1}{x_2-1}+\ldots+\frac{1}{x_n-1}$$   Source: IME entrance exam (Military Institute of Engineering, Brazil), date not provided (possibly from the 1950s)  My attempt: Developing expression $P(1)$, replacing the 1 by $x$, follows $$P(x)=\frac{(x_2-x)\cdots (x_n-x)+\ldots+(x_1-x)\cdots (x_{n-1}-x)}{(x_1-x)(x_2-x)\cdots (x_n-x)}$$    As $x_1,x_2,\ldots,x_n$ are the roots, it must be true that   $$Q(x)=(x-x_1)\cdots(x-x_n)=1+x+x^2+\ldots+x^n$$ and  $$Q(1)=(1-x_1)\cdots(1-x_n)=n+1$$ Therefore the denominator of $P(1)$ is  $$(-1)^{n} (n+1).$$ But I could not find a way to simplify the numerator.  Another fact that is probably useful is that  $$1+x^{n+1}=(1-x)(x^n+x^{n-1}+\ldots+x+1)$$ with roots that are 1 in addition of the given roots $x_1,x_2,\ldots,x_n$ for the original equation, that is  $$x_k=\text{cis}(\frac{2k\pi}{n+1}),\ \ k=1,\ldots,n.$$  This is as far as I could go...    Hints and full answers are welcomed. Are questions about (development) devices allowed on Stack Overflow? I've seen many questions on Stack Overflow asking which devices are best for developers. This made me wonder if questions asking for hardware device recommendations (best device for programming, best device within a certain budget, etc.) are on-topic for Stack Overflow, or if they should be asked somewhere else?  For Instance check these links:         etc.