So yesterday I took paper and pencil and did some constructions of geometrical nature and I discovered a sequence that should converge to $\pi$.  It goes like this:     Define $a_1=\sqrt{2}$ and for every other $n \in \mathbb N$ define $a_n=\sqrt{2+a_{n-1}}$. Define $S_n=2^n \cdot \sqrt{2-a_{n-1}}$. Then we should have $\lim_{n \to \infty}S_n=\pi$.   This formula (more correctly, sequence) can be seen  and is labeled with number (66) so I did not discover anything new.  I found this sequence but I did not rigorously prove that it converges to $\pi$, the story is that I saw a pattern for the first few natural numbers and then defined the expression for general $n$ and when I did that I found the sequence on the above linked page.  My question would be:     In which way would you prove that this sequence converges to $\pi$?
Wikipedia claims this but only cites an offline proof:  $$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+\cdots+ \sqrt 2}} = \pi$$  for $n$ square roots and one minus sign. The formula is not the "usual" one, like Taylor series or something like that, so I can't easily prove it. I wrote a little script to calculate it and it's clearly visible, but that's not a proof.
On some pages of my document LaTeX introduces a lot of spacing between my headline/paragraphs (around 5 times more than usual).  What is the reason for this and how can I avoid it? It seems it is trying to fill the page until the next chapter starts. I would like to know if it is possible to enable free shipping with Woo Commerce only if certain items (or items of certain categories / shipping classes) are included in the cart.  The Free Shipping option only activates on the basis of cart value or via a coupon code so I investigated Flat Rate shipping and discovered that it prioritises the most expensive shipping class over the cheaper ones. This means the free shipping class I created never overrides the default charge.  I understand the logic behind this but it's the opposite of what I need as the idea is to incentivise purchases of higher profit margin products by eliminating delivery charges when they're included in an order.  I don't know if what I am trying to achieve is simply not possible within Woo Commerce. A coupon that only applies to certain products / categories seems to be the closest I can get but I'd rather not force users to enter a code at checkout.  Does anybody know if there any simpler ways to give users who purchase certain products free shipping? The question is:      Prove that if $p \in \mathbb{Z}$ is irreducible, then $p$ is also prime.    Irreducible is defined as follows: $n \in \mathbb{Z} $ is irreducible iff the only way to write $n=ab$ for some integers $a$ and $b$ is if $a = \pm1$ or $b = \pm1$. Prime is defined as follows: an integer $n$ is prime iff whenever $a,b \in \mathbb{Z} $ and $p\mid ab$ then it must be true that $p\mid a$ or $p\mid b$ (or both).    Hint: Prove that if $n\mid ab$ but $n$ does not divide $a$, then $\gcd(a,n) = \pm1$. Then use Bézout's theorem and prove $n\mid b$. It's not difficult to show that  $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$  On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$.  Squaring the first power series and comparing terms gives us  $$\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}2^{-2n}=1$$  that is,  $$\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$$  My question: is there a more direct, combinatorial proof of this identity?  I've been racking my brains trying to come up with one but I'm not having much success. How can I see the phone number dialed after I initiate a call? Why does the compiler say "a constant value is required" for the first case...the second case works fine...    switch (definingGroup) {     case Properties.Settings.Default.OU_HomeOffice:           //do something           break;     case "OU=Home Office":           //do something         break;     default:         break;  }   also tried...  switch (definingGroup) {     case Properties.Settings.Default.OU_HomeOffice.ToString():           //do something         break;     case "OU=Home Office":         //do something         break;     default:         break;  }   ...same error  Here's the Properties.Setting code  [global::System.Configuration.ApplicationScopedSettingAttribute()] [global::System.Diagnostics.DebuggerNonUserCodeAttribute()] [global::System.Configuration.DefaultSettingValueAttribute("OU=Home Office")] public string OU_HomeOffice {     get {         return ((string)(this["OU_HomeOffice"]));     } } Formation of supermassive black holes Scientists have found very bright source of light which they call  and the are found to be . So these black holes are so massive that they cannot be formed by a supernova. So how are these formed? Sufficient Identification for Intra Schengen flights If one takes a direct flight from one Schengen member state to another, what's the minimum sufficient ID that needs to be provided to board the flight? How do they express the time, in American and British English? I don't know if this is a good question. But as far as I know, and as I do it, American English also say "after" other than "past" in expressing times.   For example, a quarter after six instead of, a quarter past six as in British English.  The other difference I know, so far, is that the British use dots to separate "hour" and "minute" in digital times like "08.35" while Americans use a colon e.g; "08:35".  So is what I know, so far, correct? Or are there any other differences that I don't know about? How to extract the last inserted row in SQL Server? When an insert statement is executed one or more rows is inserted into the table, is there any way to extract the last inserted row in SQL Server? How do I install PlayOnLinux on Ubuntu? Moving a very large (40 TB) DB from a 2008 server to a 2016 server I'm new here. Can anyone give me a hint in this problem? Please.  Let $X$ be a topological space. If $f,g$ are continuous function of $f,g\colon X\to S^n$  which $f(x)\neq -g(x)$ for all $x$ in $X$, show $f$ and $g$ are homotopics. I try to show a homotopy using $$q(x) = \frac{f(x)+g(x)}{\|f(x)+g(x)\|}$$ it's well defined, but i'm stuck. creating a csv file from two csv files I have two csv files:  sub_compiler.to_csv('sub_compiler.csv') sub_compiler.head()     and   sub_opt = pd.read_csv('sub_opt.csv') sub_opt.head()     and what I would like to do is to create a csv file where I have something of the form   compiler, opt   How could I do this? I need to do this to make a submission.  Thank's in advance.  [EDIT] Thank you for the answers. Now I have obtained :    don't understand why I have the columns called unnamed.   My objective is to get a csv file that contains only the columns compiler and opt. How could I do this? Thank's again.  [EDIT 2] I have solved my problem but if I open the csv file manually, so just click on the file, I have the following:    s it contains only 11 lines, but it contains 3000 lines in jupyter. Is it there something wrong?  [EDIT 3] I tried to to the following:  import pandas as pd  test = pd.read_csv('1495927.csv') test   and I have:    where 1495927.csv is the csv file I have created and that is in the image where I have 11 elements. When and how did "pretty" enter English as an intensifying adverb? Using Windows 10, I can't run certain commands from PowerShell but they all work correctly using cmd. PowerShell: node --version: node : The term 'node' is not recognized as the name of a cmdlet, function, script file, or operable program. Check the spelling of the name, or if a path was included, verify that the path is correct and try again. At line:1 char:1 + node --version + ~~~~     + CategoryInfo          : ObjectNotFound: (node:String) [], CommandNotFoundException     + FullyQualifiedErrorId : CommandNotFoundException  node.exe --version: no output. node.exe: opens a new black console (node.exe application itself). Get-Host | Select-Object Version: Version ------- 5.1.19041.906  Get-Command node: The term 'node' is not recognized as the name of a cmdlet, function, script file, or operable program. Check the spelling of the name, or if a path was included, verify that the path is correct and try again. At line:1 char:1 + Get-Command node + ~~~~~~~~~~~~~~~~ + CategoryInfo : ObjectNotFound: (node:String) [Get-Command], CommandNotFoundException + FullyQualifiedErrorId : CommandNotFoundException,Microsoft.PowerShell.Commands.GetCommandCommand Get-Command node.exe: CommandType Name Version Source ----------- ---- ------- ------ Application node.exe 14.16.1.0 C:\Program Files\nodejs\node.exe $env:Path output has nodejs &amp; git &amp; other paths. cmd: everything works fine with any command. I have git and nodejs in my PATH env variable. What's wrong with PowerShell, what a strange behavior is this and how to fix it? Evaluating $\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx$ using complex analysis I'm taking a complex analysis course and doing some practice computing residues &amp; evaluating integrals. I pulled out an old book called     On page 196-197, there are some interesting integrals to evaluate. I'm at 5.4.3.10.: I was able to do question 1, but was stumped at how to even begin with question 2:     Evaluate the integral   $$\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx ,$$   where $n$ is an natural number.   The answer is simply $\frac{2 \pi}{n}$. Any hints? How can I make schematic of this board? I want to make a schematic of this board. Can anyone tell me how? Prove that all 2x2 orthogonal matrices can be expressed as Rotation or Reflection How to format numbers as currency strings I would like to format a price in JavaScript. I'd like a function which takes a float as an argument and returns a string formatted like this: &quot;$ 2,500.00&quot;  What's the best way to do this?