If $x=1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{1000000}}$, what is the value of $\lfloor{x}\rfloor$?  I'm sure I'm supposed to use telescoping series to solve this problem, but I don't know how to start. Can someone give me a hint?
How to calculate the integer part of the value of the following equation?  $$y=1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+\ldots+\frac{1}{\sqrt{1000000}}$$  It should be calculated in a special way, after all the equation is so long.
I got stuck on this inequality for a day. If $p$ is positive integer, then the problem becomes too easy, but I can't find how we deal with the general case when p can be any positive number. Can someone give me a hint? I really appreciate     Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$ To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$ Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations  $(b+c+d)^{2010}=3a$  $(a+c+d)^{2010}=3b$  $(a+b+d)^{2010}=3c$  $(a+b+c)^{2010}=3d$  I tried to find the solutions using hit and trial and by using some logic also I find two solutions which are $(0,0,0,0)$ and $(\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3})$  I think these are the only solutions that exist. I noticed that when I had a helium filled, latex balloon inside of my car, it moved forward in the cabin as I accelerated forward. The faster I accelerated forward, the faster the balloon went from the back of the car to the front of the car. The balloon didn't have a string.  This became a game with my 4 year old as we drove home. We figured out where the balloon would go based on how fast I accelerated, turned corners etc.  I expected that it would act a lot like the water in a cup does, but it was the total opposite it seemed.  What forces caused this behavior?  I assumed it has something to do with the fluid dynamics in the closed cabin, but I can't figure it out. update line based on previous line Elementary central binomial coefficient estimates How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}&lt;\binom{2n}{n}&lt;\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ &gt; 1 ?     Does anyone know any better elementary estimates?   Attempt.  We have $$\frac1{2^n}\binom{2n}{n}=\prod_{k=0}^{n-1}\frac{2n-k}{2(n-k)}=\prod_{k=0}^{n-1}\left(1+\frac{k}{2(n-k)}\right).$$ Then we have $$\left(1+\frac{k}{2(n-k)}\right)&gt;\sqrt{1+\frac{k}{n-k}}=\frac{\sqrt{n}}{\sqrt{n-k}}.$$ So maybe, for the lower bound, we have $$\frac{n^{\frac{n}{2}}}{\sqrt{n!}}=\prod_{k=0}^{n-1}\frac{\sqrt{n}}{\sqrt{n-k}}&gt;\frac{2^n}{\sqrt{4n}}.$$ By Stirling, $n!\approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$, so the lhs becomes $$\frac{e^{\frac{n}{2}}}{(2\pi n)^{\frac14}},$$ but this isn't $&gt;\frac{2^n}{\sqrt{4n}}$. Where to put the periods when using a parenthetical sentence? I have two complete sentences; the second is in parentheses, essentially a comment on the first.  Where do I place the periods?  I would use this:     They played opera. (I covered my ears.)   Is this punctuation correct? I have Java 8 and Ubuntu 18.04. I'm using NetBeans and when tried to make some program, it couldn't be done because JavaFX is missing.  I installed Java JDK with NetBeans from Oracle's site and thought everything is included but obviously I didn't check it well.   What should I do, what should I install? I'm not so good at this and don't know if problem is with Ubuntu or Java. :( Many SE sites (like math.SE) have a problem with mixed content, which causes display bugs in recent browsers. This is particularly evident with math.SE (because of the huge use of MathJax) and with chat sections.  Will this be fixed?  How to reproduce  Visit  or   What happens  Mixed content is blocked, output is missing pieces of code (MathJax):    This picture also includes my main Firefox addons (top right), it relevant. It shows missing styles:    The white on the left is my tab list. Baking with oven without bottom heating element My mom wants to do baking but her oven doesn't have bottom heating element. It only has an element on top, with a fan at the back.  She told me that she can't make me a good pizza or cake without it.  Is there any way around? How to handle a native OS (Ubuntu) file upload window in selenium webdriver with python? This is my first time here but I have already posted this question on SO. I keep getting solutions for Windows + Java and not for Linux or python for that matter.  I am using Selenium and Python to automate a file upload application for the past many hours but nothing helpful!   The HTML DOM structure for this file upload is an input tag with its type attribute value = 'file'. I know I can make use of the send_keys.   The problem is I am not supposed to use send_keys() (as per the question I am trying to solve). The assumption is there are other ways of doing it.  Can anyone suggest me alternative ways to upload the file? I do get solutions for windows but not for Linux(Ubuntu). I know there are applications/tools to automate GUI components like AutoIt but there is nothing for Linux close to AutoIt.  I can say I have done my homework to find a solution. Only after not finding anything useful, I am asking this question, please help! Can you get a Windows (AD) username in PHP? I have a PHP web application on an intranet that can extract the IP and host name of the current user on that page, but I was wondering if there is a way to get/extract their Active Directory/Windows username as well. Is this possible? How to say that the phone is held between shoulder and ear? What are common phrases an English native speaker would use for the pose of speaking on the phone where you use your shoulder to keep the receiver against your ear (and mouth), so your hands are free?  I have some ideas, but they all sound wooden, and I don't think I've ever heard anyone describe this pose in words. It's also pretty much impossible to google for a pose or look it up on leo.org or other dictionaries.  OK, since people keep asking for what phrasings I found, I'll provide two, though I was hoping I could avoid influencing people who answer here by not providing them: "balance on one's shoulder", "held/nestled/sandwiched between shoulder and ear". Traditional search-as-you-type on newer Nautilus versions Until now I was using a "traditional" version of Nautilus: that that comes with Ubuntu 10.10. Now I'm using 3.6.3, included with Ubuntu 13.04 (most likely any recent version is equivalent for the scope of my question, anyway).  One major difference between the two is the search-as-you-type behavior: in classic versions of Nautilus, when you typed an input, the file with the closest alphabetical match would become the selected one, the eligible files/folders being restricted to the current directory.  The new behavior is to perform an optionally-global search instead.  Can one configure Nautilus to provide the previous functionality ? Suspended user cannot award bounty Quote unquote (or end-quote), unseparated by the actual quotation In spoken English, people often say "quote-unquote" (or "quote-endquote") to indicate that part of what they are saying is a quotation (scare or otherwise). Sometimes the quoted material will go between quote and unquote, and occasionaly it will precede the quote-unquote. Often, though, "quote unquote" introduces the quoted material.  Why does quote unquote precede the quotation?  It would seem that quote opens the quoted text and that unquote closes it, but then why would we close the quotation before actually providing it. It would be the equivalent of writing your quotation marks together, "", rather than around the quoted text.  What are the origins of this construction?  Three years ago ChrisR asked  on whether it is correct to use quote, unquote. The answers did not deal with origins, but one commenter suggested that quote-unquote is a corruption of quote-on-quote, but he provided no evidence, nor any real explanation. Iterating through a Collection, avoiding ConcurrentModificationException when removing objects in a loop What does "No bootable medium found" mean in VirtualBox? How can I get bold math symbols? To make Latin-letter variables bold I can use e.g. \mathbf{a}, but while putting Greek letters or symbols such as \nabla inside \mathbf doesn't cause any errors or warnings, it also doesn't do anything else.  What is the best way to make bold math symbols, in particular Greek letters and \nabla? How do I perform a Perl substitution on a string while keeping the original? In Perl, what is a good way to perform a replacement on a string using a regular expression and store the value in a different variable, without changing the original?  I usually just copy the string to a new variable then bind it to the s/// regex that does the replacement on the new string, but I was wondering if there is a better way to do this?  $newstring = $oldstring; $newstring =~ s/foo/bar/g;