How to find the value of the infnite product  $\frac {7}{9}$x$\frac {26} {28}$x $\frac {63} {65}$x . . . x$\frac {n^3-1} {n^3+1}$x . .  Somebody please help me.
How to compute   $$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}\ ?$$   My Working :   $$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}= 1 - \prod^{\infty}_{n=2}\frac{2}{n^3+1} =  1-0 = 1$$  Is it correct
Simple way to see a list of installed iOS apps with their storage space size? Consciousness is an electrical and chemical interaction in the brain, caused by neurons firing and chemical interactions. How does a mechanical "force" cause this to stop working?  i.e. How does a mechanical action (such as a punch to the head) cause this electrical/chemical system to stop working (leading to unconsciousness)? Average length of the longest segment This post is related to a previous SE post  concerning average length of a smallest segment.  A rope of 1m is divided into three pieces by two random points. Find the average length of the largest segment. My answer is 11/18.  Here is how I do it:  Here we have two independent random variables $X,Y$, both uniform on $[0,1]$. Let  $A=\min (X,Y), B=\max (X,Y)$ and $C=\max (A, 1-B, B-A)$. First we want to find the probability density function $f_C(a)$ of $C$. Let $F_C(a)$ be the cumulative distribution function. Then $$ F_C(a) = P(C\le a)=P(A\le a, 1-B\le a, B-A\le a).$$  By rewriting this probability as area in the unit square, I get $$F_C(a)=\left\{\begin{array}{ll} (3a-1)^2 &amp; \frac{1}{3}\le a\le \frac{1}{2}\\ 1-3(1-a)^2 &amp; \frac{1}{2}\le a\le 1\end{array}\right.$$ from which it follows that  $$f_C(a)=\left\{\begin{array}{ll} 6(3a-1) &amp; \frac{1}{3}\le a\le \frac{1}{2}\\ 6(1-a) &amp; \frac{1}{2}\le a\le 1\end{array}\right.$$ Therefore the expected value of $C$ is $$\int_{1/3} ^{1/2}6a(3a-1) da+\int_{1/2} ^{1}6a(1-a) da= \frac{11}{18}.$$  My questions are:  (A) Is there a "clever" way to figure out this number 11/18?  (B) What is the answer if the rope is divided into $n&gt;3$ segments? Let $y_n$ be a sequence such that  $\lim_{n\to \infty} y_n= y$ then prove that     $ \lim \dfrac{y_1+y_2+...+y_n}{n}= y$   My try:  As  $\lim_{n\to \infty} y_n= y$ then given $\epsilon&gt;0$ there exists $N$ such that $\forall n\ge N ;||y_n-y||&lt;\epsilon $.  Now $||\dfrac{y_1+y_2+...+y_n}{n}-y||\le n^{-1}\{||y_1-y||+||y_2-y||+...+||y_n-y||\}$  How to make this limit tend to zero? What is "%AppData%"? How do I find out the user name and password for an existing account? Variational Derivation of Schrodinger Equation In case of a syntax error, pdflatex usually (in \errorstopmode) gets interactive and asks you what to do. I know I can press:   X to terminate S to enter \scrollmode (continue processing without stopping if further syntax errors arise, but stop for other errors) R to enter \nonstopmode (continue processing without stopping if further errors arise) Q to enter \batchmode (continue processing without stopping if further errors arise and suppress further output to the terminal)   Apparently, only the latter three (SRQ) can be provided implicitly by the respecive \...mode commands or the -interaction=...mode command line option. However, what I need is a mode that mimics the implicit pressing of X, that is, to make the pdflatex process terminate after the first error. How to use HttpWebRequest (.NET) asynchronously? How can I use HttpWebRequest (.NET, C#) asynchronously? Few application icons not appearing in Ubuntu Dock after upgrading to Ubuntu 17.10 Computing the best constant in classical Hardy's inequality Classical Hardy's inequality (cfr. Hardy-Littlewood-Polya Inequalities, Theorem 327)  If $p&gt;1$, $f(x) \ge 0$ and $F(x)=\int_0^xf(y)\, dy$ then   $$\tag{H} \int_0^\infty \left(\frac{F(x)}{x}\right)^p\, dx &lt; C\int_0^\infty (f(x))^p\, dx $$  unless $f \equiv 0$. The best possibile constant is $C=\left(\frac{p}{p-1}\right)^p$.   I would like to prove the statement in italic regarding the best constant. As already noted by Will Jagy , the book suggests stress-testing the inequality with   $$f(x)=\begin{cases} 0 &amp; 0\le x &lt;1 \\ x^{-\alpha} &amp; 1\le x \end{cases}$$   with $1/p&lt; \alpha &lt; 1$, then have $\alpha \to 1/p$. If I do so I get for $C$ the lower bound  $$\operatorname{lim sup}_{\alpha \to 1/p}\frac{\alpha p -1}{(1-\alpha)^p}\int_1^\infty (x^{-\alpha}-x^{-1})^p\, dx\le C$$  but now I find myself in trouble in computing that lim sup. Can someone lend me a hand, please?    UPDATE: A first attempt, based on an idea by Davide Giraudo, unfortunately failed. Davide pointed out that the claim would easily follow from   $$\tag{!!} \left\lvert \int_1^\infty (x^{-\alpha}-x^{-1})^p\, dx - \int_1^\infty x^{-\alpha p }\, dx\right\rvert \to 0\quad \text{as}\ \alpha \to 1/p. $$  But this is false in general: for example if $p=2$ we get   $$\int_1^\infty (x^{-2\alpha} -x^{-2\alpha} + 2x^{-\alpha-1}-x^{-2})\, dx \to \int_1^\infty(2x^{-3/2}-x^{-2})\, dx \ne 0.$$ I have an external hard disk, and for some reason, it is reporting wrong free space.  In Windows Explorer, it is showing the used space is 378 GB.    I tried running chkdsk H: /f /r, it seems to report the same usage as Windows Explorer, and it says that there is no problem.    But when I tried selecting all the files inside this drive, it is showing much lesser usage.    I've already set the setting to show all hidden files and folders, also show the protected OS files, but it still cannot find about those missing files. And the difference is huge (282 GB). I've already cleaned the Recycle bin as well.    What I am missing here and how do I identify those missing spaces?  [UPDATE]: I used  to help identify and it seems to find some gigantic files, of which filename is quite weird. Anyone knows what this is and is it safe to delete? Is there a way to use the covariance matrix to find coefficients for multiple regression? For simple linear regression, the regression coefficient is calculable directly from the variance-covariance matrix $C$, by  $$ C_{d, e}\over C_{e,e} $$ where $d$ is the dependent variable's index, and $e$ is the explanatory variable's index.  If one only has the covariance matrix, is it possible to calculate the coefficients for a model with multiple explanatory variables?   ETA: For two explanatory variables, it appears that $$\beta_1 = \frac{Cov(y,x_1)var(x_2) - Cov(y,x_2)Cov(x_1,x_2)}{var(x_1)var(x_2) - Cov(x_1,x_2)^2} $$ and analogously for $\beta_2$. I'm not immediately seeing how to extend this to three or more variables. System.CalloutException: You have uncommitted work pending I have same exception can please   The previous post issue is resolved.  Exception: :System.CalloutException: You have uncommitted work pending. Please commit or rollback before calling out  Visualforce Page   &lt;apex:page controller="Controller" action="{!init}" showHeader="false" sidebar="false"&gt;     &lt;center&gt;         &lt;apex:pageBlock title="Request Listener"&gt;&lt;/apex:pageBlock&gt;               &lt;/center&gt; &lt;/apex:page&gt;   Controller :            PUBLIC with sharing class Controller {        PUBLIC String fromNumber      = ApexPages.currentPage().getParameters().get('From');     PUBLIC String toNumber        = ApexPages.currentPage().getParameters().get('To');     PUBLIC String body            = ApexPages.currentPage().getParameters().get('Body');        PUBLIC PageReference init()     {                    System.debug('From Phone Number :' +fromNumber);         System.debug('To phone NUmber :' + toNumber);         System.debug('Message Body :' + body);          TwilioRestClient Client = TwilioAPI.getDefaultClient();                                                                SYSTEM.DEBUG('FROM AND To  Number is NOT NULL');                      String formattedNumber='+919876543210';                 IF(body != NULL)                 body = body;             ELSE                 body = '';             Case c = NEW Case (Subject = formattedNumber,Description = body,Origin = 'Phone');             INSERT c;                               Map&lt;String,String&gt; params1 = new Map&lt;String,String&gt;             {                'To'   =&gt; fromNumber,                'From' =&gt; '+1908280****',                'Body' =&gt; 'Conformation to customer'             };             TwilioMessage message = client.getAccount().getMessages().create(params1); /* Valid conformation SMS sent to the Customer.*/                                            return null ;     } }   While loin as admin then click preview on visualforce page executed but case is not created.I have checked the the debug log it throw the exception. What is an intuitive explanation for how PCA turns from a geometric problem (with distances) to a linear algebra problem (with eigenvectors)? I've read a lot about PCA, including various tutorials and questions (such as , , , and ).   The geometric problem that PCA is trying to optimize is clear to me: PCA tries to find the first principal component by minimizing the reconstruction (projection) error, which simultaneously maximizes the variance of the projected data.    When I first read that, I immediately thought of something like linear regression; maybe you can solve it using gradient descent if needed.  However, then my mind was blown when I read that the optimization problem is solved by using linear algebra and finding eigenvectors and eigenvalues. I simply do not understand how this use of linear algebra comes into play.  So my question is: How can PCA turn from a geometric optimization problem to a linear algebra problem? Can someone provide an intuitive explanation?  I am not looking for an answer like  that says "When you solve the mathematical problem of PCA, it ends up being equivalent to finding the eigenvalues and eigenvectors of the covariance matrix." Please explain why eigenvectors come out to be the principal components and why the eigenvalues come out to be variance of the data projected onto them  I am a software engineer and not a mathematician, by the way.  Note: the figure above was taken and modified from . My stance is that these websites are a great boon to the research community and they were certainly extremely helpful for my research. I think they should be applauded rather than persecuted, and while they may be illegal now, I think a way to make them legal should be found (in the same way I can download articles legally through my university's subscription to all the major publishers and journals).  Can I thank them in the acknowledgements section of my thesis? I would like to know how to put a time delay in a Python script. $a_1 a_1', a_2 a_2', ..., a_n a_n'$ is not a reduced modulo system $\mod p$. If $a_1, a_2, ..., a_n$ is a reduced residue system $\mod p$ and $a_1', a_2', ..., a_n'$ is another reduced residue system $\mod p$ then $a_1 a_1', a_2 a_2', ..., a_n a_n'$  is not a reduced modulo system $\mod p$.  Not able to show a counter to it. Help Needed. How can I fix a 404 Error when using a PPA or updating my package lists? PPA seems to be constantly offline. Whenever I use sudo apt-get update, this error is shown:  W: Failed to fetch http://ppa.launchpad.net/ 404  Not Found   How do I fix these errors? How do I change the default application for unknown file types? Ever since installing Bless Hex Editor, all unknown files and text files automatically open with Bless instead of gedit. How do I change it back? I don't want to have to manually associate files with gedit, I just want gedit to be the default if there's no other application set up to open the file.  Solution: modify ~/.local/share/applications/mimeapps.list  [Default Applications]  - text/plain=bless.desktop - application/octet-stream=bless.desktop + application/octet-stream=gedit.desktop