I found this problem and according to Wolfram Alpha, the answer is 2.  $$\prod_{n=0}^{\infty} \left (1+\frac{1}{2^{2^n}}\right)$$  Please do the favor of explaining me how the product is 2.
What will be the value of the following Infinite Product :   $$\displaystyle \prod_{k=0}^\infty \left(1+\dfrac{1}{2^{2^k}}\right)$$  It would be nice if anyone could spare the time and boil down to the absolute basics and tell how they reached the solution.
Need rtl8814au driver for kernel 5.3 on Ubuntu 19.10 Proof about Graph with no triangle I am using \if condition in a custom command in my LaTeX document. I would like to know where I can find some good explanation about \if command. I followed a  but it's just an example; there are not any explanation. How to run a complete GUI as root? I don't care if I mess up the system (which is likely if I do things like this), or the potential security issues that this could imply. Note, that I don't just want to "", but run my GUI from bottom-up as root. I don't want to type my password for anything but just for login and have total control over it.  Of course, I want any application running as root, like Firefox, LibreOffice, etc. without having to mess up with the terminal. So, how do I do this?     This question tries to address users that do not like the current permission system with root being separated from normal users. I do not endorse such behaviors, but if a user wants to, for whatever reason, this question should fit their corncerns.   Not advisable for new users I'm trying to write a small script to change the current directory to my project directory:  #!/bin/bash cd /home/tree/projects/java   I saved this file as proj, added execute permission with chmod, and copied it to /usr/bin. When I call it by: proj, it does nothing. What am I doing wrong? What is the correct way to create a single-instance WPF application? Using C# and WPF under .NET (rather than  or console), what is the correct way to create an application that can only be run as a single instance?  I know it has something to do with some mythical thing called a mutex, rarely can I find someone that bothers to stop and explain what one of these are.  The code needs to also inform the already-running instance that the user tried to start a second one, and maybe also pass any command-line arguments if any existed. After logging in on terminal with user name and password, how do I start the GUI? Which objects can I use in a finalizer method? copy a certain range of one sheet to a certain specifed cell in other sheet show all my unanswered questions from SE Any way how to get    my all un-answered questions  from across all SE sites? How can I link to an external resource in a community-friendly way? I would like to help my fellow Stackizens, using material from my blog or other website.  But external links are sometimes viewed as excessive promotion.  From the FAQ:     Be careful, because the community   frowns on overt self-promotion and   tends to vote it down and flag it as   spam. Post good, relevant answers, and   if they happen to be about your   product or website, so be it. However,   you must disclose your affiliation in   your answers. Also, if a huge   percentage of your posts include a   mention of your product or website,   you're probably here for the wrong   reasons.   I have tried posting answers that reference my blog, and I have followed the above rules, but my answers still get flagged as spam.  How can I craft an answer that references my blog, website or other resource, but is still acceptable to the community? android how an EditText work as AutoComplete how to get a picklist all values in apex controller I am wondering if the mass density profile $\rho(\vec{r})$ has been characterized for atomic particles such as quarks and electrons. I am currently taking an intro class in quantum mechanics, and I have run this question by several professors. It is my understanding from the viewpoint of quantum physics a particle's position is given by a probability density function $\Psi(\vec{r},t)$. I also understand that when books quote the "radius" of an electron they are typically referring to some approximate range into which an electron is "likely" to fall, say, one standard deviation from the expectation value of its position or maybe $10^{-15}$ meters.  However it is my impression that, in this viewpoint, wherever the particle "is" or even whether or not the particle "had" any position to begin with (via the Bell Inequalities), it is assumed that if it were (somehow) found, it would be a point mass. This has been verified by my professors and GSIs. I am wondering if its really true.  If the particle was truly a point mass then wherever it is, it would presumably have an infinite mass density. Wouldn't that make electrons and quarks indistinguishable from very tiny black holes? Is there any practical difference between saying that subatomic particles are black holes and that they are point masses? 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Reverse (parse the output) of Arrays.toString(int[]) Is there in the JDK or Jakarta Commons (or anywhere else) a method that can parse the output of Arrays.toString, at least for integer arrays?  int[] i = fromString(Arrays.toString(new int[] { 1, 2, 3} ); Do you pluralize the singular possessions of / items or people associated with individual members of a plural group? In a situation where say a group (or at least a plurality) of men is being addressed — for example on a sign passed by many married men — which is correct?  &quot;Remind your wife.&quot;  or  &quot;Remind your wives.&quot; Unzip a file with php