Let we have the following sequence $$x_1=1$$ $$x_{n+1}=\frac{x_n^2+a}{2x_n}$$ such that $$a&gt;0$$ how can I prove it is a convergent sequence and its limit is $\sqrt{a}$
a) Let $a&gt;0$ and the sequence $x_n$ fulfills $x_1&gt;0$ and $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$ for $n \in \mathbb N$. Show that $x_n \rightarrow \sqrt a$ when $n\rightarrow \infty$. I have done it in two ways, but I guess I'm not allowed to use the first one and the second one is incomplete. Can someone please help me?  We already know $x_n \rightarrow \sqrt a$, so we do another step of the iteration and see that $x_{n+1} = \sqrt a$.  Using limit, $x_n \rightarrow x, x_{n+1} \rightarrow x$ (this is the part I think it's incomplete, don't I have to show $x_{n+1} \rightarrow x$, how?), we have that   $$x = \frac x 2 (1 + \frac a {x^2}) \Rightarrow 1 = a/x^2 \Rightarrow x = \sqrt a$$ b) Let the sequence $x_n$ be defined as $x_{n+1}= 1 + \frac 1 {x_n} (n \in \mathbb N), x_1=1$. Show that it converges and calculate its limit. &quot;Tip: Show that sequences $x_{2n}$ and $x_{2n+1}$ monotone convergent to the limit.&quot; I didn't understand the tip, how can this help me? Does it make a difference if the number is odd or even? Thanks in advance!
Avoiding IF ELSE due to variable been NULL Why is Excel truncating my 16-digit numbers? If I type in a 16 digit number (format: number, no decimals) it changes the number on me. Example: 1234567812345678 changes the view to 1234567812345670.    If I type it in as a general format it changes the numbers above so it displays 1.23457E+15 but if you click on the cell, the display shows the last digit as a 0 instead of an 8 once again.   I opened the file on a different computer and same issue now with it.  I have changed the auto correction and auto formatting all to no avail.  Help! Lightning Web Component embedded in Visualforce Page via Aura Component and App not displaying I am writing a lightning web component that is going to be used in Salesforce Classic, Lightning Experience and an Employee Community. It does not HAVE to be a lightning web component - I could use a traditional Aura Lightning Component if I must.   However, what I am finding is that the visualforce page does not work, even when the lwc component is embedded in a aura component that is then used with Lightning Components for Visualforce. I can see from the documentation  that using a lwc directly from within visualforce is not supported, but I was under the impression that it may work if wrapped in an aura component.   My question then is this:  Has anyone had success having a lwc wrapped in an aura component display successfully within Visualforce? Or is this completely unsupported behaviour? Iridescence In Cycles? Is it possible to create an  shader in Cycles? Specifically for something like , , , etc.. Since , how do I uninstall it from my desktop system?  (Please note: This question is not about Unity 7, which is still supported at the time of this writing and should remain installed if present.) Credit score just tanked ~50 points and I'm not sure why I check my credit score monthly using a credit tracking app provided by my credit card company. I use a reminder to check on the same day each month, and have been tracking my credit score for almost two years.  In the last year, the app switched from a TransUnion model to VantageScore 3.0. I have TransUnion data for the entire duration, but only about a year's worth of VantageScore data.  I have good credit, and my TransUnion score has increased steadily from ~740 to ~780 in the last two years.  My VantageScore started around ~760 and increased to ~800 last month. This morning when I checked my credit score, it had dropped just over 50 points to ~750.  My TransUnion score has stayed solid at ~780.  My questions are:  Should I be concerned about this sudden change in score?  Is there any way I can determine what changed to account for the sudden change in credit score? Uses of the 3D Cursor I know the 3D Cursor can be used to position new objects, or snap existing ones to a certain location - but what else can it be used for?  An exhaustive list of every possible use would be great (without repeating similar things like "pivot for spin tool", "pivot for screw tool" - which could simply be said as "Pivot for modeling tools"). Why does Property Set throw StackOverflow exception? I know java and would normally put in getter/setter methods. I am interested in doing it in C# with the following code, but it throws a StackOverflow exception. What am I doing wrong?  Calling Code  c.firstName = "a";   Property Code    public String firstName; {     get     {         return firstName;     }     set     {         firstName = value;     } } How do I get the size of a directory on the command line? Does the Darkness spell block vision? In discussing , I came to realize that the real question was whether Darkness blocks vision or merely creates "darkness".    Per the wording of the spell:     Magical darkness spreads from a point you choose within range to fill a 15-foot-radius sphere for the duration. The darkness spreads around corners. A creature with darkvision can’t see through this darkness, and nonmagical light can’t illuminate it. (emphasis mine)   Normal darkness is defined in the game as creating a heavily obscured area and the only description added to the darkness is that it is magical, which (to me) just means that it is created by magic and is subject to the magical rules.    Darkvision is defined in the game as basically being able to see in darkness as if they were seeing in dim light. So, the text "darkvision can’t see through this darkness" merely means that it affects darkvision in the same way it affects normal vision. (See earlier versions of the spell below, which had similar wordings.)    I see nothing there that implies it is a barrier to vision, just an active  and utter absence of illumination.    Previous editions have varying descriptions.  and  give accurate accountings of the various versions.   The 1e Wizard version seems to actually imply a sphere of opaque blackness that even blocks infra/ultravision, while the 1e Cleric version is a reversal of the Light spell and creates totally normal darkness, with no block to special visions.   The 2e version is similar to the 1e Cleric spell. It creates an area of darkness "equal to an unlit interior room".   The 3.0e version comes closest to the 5e version. It "causes an object   to radiate darkness out to a 20-foot radius. Not even creatures who can normally see in the dark (such as with darkvision) can see in an area shrouded in magical darkness." Still no reference to any sort of opacity and the wording makes it clear that darkvision is being treated specifically here.   3.5e Darkness is similar, although instead of darkness, it creates "shadowy illumination". This affects darkvision as well as normal vision.   4e doesn't have a Darkness spell, but rather a Cleric Utility called Veil of Darkness, which creates "a zone that is heavily obscured and blocks line of sight." So, finally, a reference to opacity, in "blocks line of sight", although I question if opacity was the intent. Addons: Only some .zips will install Why is it that if I go to install an add-on from User Preferences > Add-ons > Install from File... and choose a *.ZIP, some add-ons will install, while others will return me to the User Preferences with nothing having happened (as far as I can tell)?  Is there a specific file structure required in the .ZIP? I get many add-ons from GitHub, using the Download Zip button, and even there installation of the actual Zip file only works some of the time. We've heard it many times, nothing can escape the gravity of a black hole, even light once it's past the event horizon.  If this is true, how can a black hole emit anything?  are massive black holes that have consumed enormous amounts of matter and energy, and then blasted it back out into the universe.  How is this possible?  Time stands still in a black hole, that in itself should stop anything from moving. I have a class like this:  public class myClass {   public List&lt;myOtherClass&gt; anewlist = new List&lt;myOtherClass&gt;;    public void addToList(myOtherClass tmp)   {     anewList.Add(tmp);   }  }   So I call "addToList" a hundred times, each adding a unique item to the list.  I've tested my items to show that before I run the "addToList" method, they are unique.  I even put a line in to test "tmp" to make sure it was what I was expecting.  However, when I do this (lets say myClass object is called tmpClass):  int i = tmpClass.anewList.Count(); for (int j = 0; j&lt;i; j++) {    //write out each member of the list based on index j... }   I get the same exact item, and it's the last one that was written into my list.  It's as if when I add, I'm overwriting the entire list with the last item I've added.    Help?  This makes no sense. I've also tried List.Insert, where I'm always inserting at the end or at index 0.  Still no dice.  Yes, I'm doubly source my indexing is correct and when I do my test I'm indexing through each of the elements.  :)  UPDATE: Okay, I tried this and still had the same problem:  foreach(myOtherClass tmpC in tmpClass.anewList) {         Console.WriteLine(tmpC.theStringInMyClass.ToString()); }   and still for each of the 100 items, I got the same string output... I'm sure I'm doing something completely stupid, but I don't know what yet. I'm still 100% sure that the right string is getting passed in to begin with.  -Adeena    Okay, I tried this and still had the same problem:  foreach(myOtherClass tmpC in tmpClass.anewList) {     Console.WriteLine(tmpC.theStringInMyClass.ToString()); }   and still for each of the 100 items, I got the same string output...  I'm sure I'm doing something completely stupid, but I don't know what yet. I'm still 100% sure that the right string is getting passed in to begin with.  -Adeena How can I modify title page so that it looks like it is in  video. Please seek to 16 seconds and watch title page. I do not have permission to post images. I liked this theme and I want to make a beamer theme that looks similar. I tried with beamercolorbox but I failed to position the box to the bottom.      Edit: Here is my minimum working example  \documentclass[xcolor=dvipsnames]{beamer} \defbeamertemplate*{title page}{customized}[1][] { \usebeamerfont{subtitle} \usebeamercolor[fg]{subtitle}  {\flushright \setbeamercolor{author}{bg=white,fg=Red} \usebeamerfont{author} \insertauthor\par \usebeamerfont{institute}\insertinstitute\par }  \vspace{0.2in}  \setbeamercolor{postit}{fg=white,bg=YellowOrange}  \begin{beamercolorbox}[sep=1em,wd=1.062\textwidth,ht=3cm,dp=3cm,right]{postit} \usebeamerfont{title} {\huge \inserttitle}\par \usebeamerfont{subtitle} {\insertsubtitle} \end{beamercolorbox} } \usecolortheme[named=YellowOrange]{structure} \useoutertheme{infolines} \setbeamertemplate{navigation symbols}{}  \institute{Harvard College Professor and\\ Johnstone Family Professor in Psychology, Harvard University}  \title{SAY WHAT?} \subtitle{Linguistics as a Window to\\ Understanding the Brain} \author{Steven Pinker}  \begin{document} { \setbeamertemplate{footline}{} \begin{frame} \titlepage \end{frame} } \end{document} My book uses the argument that the multiplicities of a few macrostates in a macroscopic object take up an extraordinarily large share of all possible microstates, such that even over the entire lifetime of the universe, if each microstate had an equal chance of being accessed, fluctuations away from equilibrium would never occur.  My question to this statistical proof is this: In the real world, is there really an infinitesimal but nonzero chance that macroscopic systems could access some of the more unlikely macrostates (e.g. heat transfer from a cold object to a hot object)? How can I get fireworks? What are the requirements for making fireworks appear at the end of a level in the original Super Mario Bros.? Show that $\mathbb{Q}(\sqrt{2})$ is the smallest subfield of $\mathbb{C}$ that contains $\sqrt{2}$ This was an assertion made in our textbook but I have no idea how to show that either statement is true.  Also would like to show that that $\mathbb{Q}(\sqrt{2})$ is strictly larger than $\mathbb{Q}$, which was the second part of the assertion. I am using lightning:recordEditForm and lightning:inputField for creating a new record. this component is being used inside a lightning community builder. When i click the submit button, i am able to create a new record in my custom object. However i need the created record ID as i need to pass it in the URL for the next page. My dad's old TV! When I switch it off after watching and get my hands closer, my hands' hair gets attached to the screen. And they stick for sometimes! Now, I know how a cathode ray oscilloscope works. It is just the clever use of Hall - Effect . But why does my hair get stuck to the screen? What is the cause? Note, I checked this for my LED TV but it didn't work (I couldn't perceive anything) unknowingly. What's the magic in my old TV?? According to the Wikipedia article on  the sample SD   $$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2}$$  is a biased estimator of the SD of the population. It states that $E(\sqrt{s^2}) \neq \sqrt{E(s^2)}$.   NB. Random variables are independent and each $x_{i} \sim N(\mu,\sigma^{2})$   My question is two-fold:   What is the proof of the biasedness? How does one compute the expectation of the sample standard deviation    My knowledge of maths/stats is only intermediate.