Given $S_{1}=2$ and $S_{n+1}= \frac{S_{n}}{2} + \frac{1}{S_{n}}$. Assume $ S_{n} > 1$, show that $S_{n+1} > 1$
Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$
In order to protect a portal in the nether I have an obsidian room with an iron door, which works off a switch in the room. The problem is that if I leave the room and someone else on the server closes the door from the inside I can't get back to that portal.  Powering red stone on the outside of the door doesn't seem to work, as the switch inside is already powering it, so I'm guessing something like an XOR gate will be the solution, however I am finding it difficult to see how to make this work in practice. How do I install the latest version of Matlab? When I tried to install Matlab using the Software Center, it prompted me for the Location of MATLAB installation(s). Since I am trying to install it, I don't know what to do. xgboost tayler expansion detail This is the objective function for Xgboost.  I have no idea where $g_{i}$ and $h_{i}$ came from  is some one explain how this two terms came form?  or direct me to the related tutorial page then I will search for it Make a momentary switch control a toggle What are the simplest, cheapest, smallest ways to make a momentary switch produce a 2-state toggling output (latching momentary switch)?  In other words, the output is continuously low, and when you momentarily press the button/tact switch, the output changes to continuously high, and then when you press it again, it switches back to low. C++ warning: deprecated conversion from string constant to ‘char*’ [-Wwrite-strings] What sort of cheat sheets for a new DM lead to a better flowing 5e game of Dungeons and Dragons? The question is, what elements on a DM screen are essential? I'm new at playing and DMing, altough a lot of experience watching games, just got a group to play with now. Started with the starter set in 5E and the Lost Mine of Phandelver adventure, however I've noticed the game can come to a stop when checking the rules or how to do something, while getting the books and stuff like that. So I thought to make a DM screen with some cheat sheets, of the most usual things new players, and new DM's have doubts on, and how to make the game flow well without long rule-checking breaks. What are the essential things to have on that screen? So far this is what I found  but not sure if it has all I need or so much stuff that'll also make the game stop a lot while cheking it out. "Like" versus "as" "Stay home" or "stay at home" – which is correct and why? Bridged interfaces and VLAN tags I want Ctrl+Alt+B bring up a browser.  Ctrl+Alt+Delete is handy.  Can I set up Ctrl+Alt+B to bring up a browser (either default or specific)? Let $A \in {M_n}$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$? If $Y|X \sim \mathcal{N}(0,1)$ then is $Y^2|X \sim \chi^2(1)$? Suppose we have random variables $X$ and $Y$ such that $Y|X \sim \mathcal{N}(0,1)$. Can we then say that $Y^2|X \sim \chi^2(1)$? If we can, then what about when $Y|X \sim \mathcal{N}(0,\sigma^2/4)$, can we use the results of  to say that $Y^2|X \sim \frac{\sigma^2}{4}\chi^2(1)$? Essentially I want to know if the results of squaring a normal distribution can be applied to the case where a random variable is conditionally normally distributed? See below in acute triangle $ABC$, $D$ is on $AC$ such that $AD=BC$. $CF$ is the angle bisector of $\angle ACB$. $DE \parallel CF$. $E$ is on $AB$. $AE = CD$.  Prove that $\angle ABC = 2 \angle BAC$. In addition, find the condition in which $\angle ADB = 3 \angle BAC$  I struggle to find ways to use two conditions $AD = BC$ and $AE = CD$ When I change the stroke width of a rectangle the width and and height change also. How can I  prevent this from happening? Proving that $\lim\limits_{x\to\infty}f'(x) = 0$ when $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ exist I've been trying to solve the following problem:  Suppose that $f$ and $f&#39;$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to\infty}f&#39;(x)$ exist. Show that $\displaystyle\lim_{x\to\infty}f&#39;(x) = 0$.  I'm not entirely sure what to do. Since there's not a lot of information given, I guess there isn't very much one can do. I tried using the definition of the derivative and showing that it went to $0$ as $x$ went to $\infty$ but that didn't really work out. Now I'm thinking I should assume $\displaystyle\lim_{x\to\infty}f&#39;(x) = L \neq 0$ and try to get a contradiction, but I'm not sure where the contradiction would come from.   Could somebody point me in the right direction (e.g. a certain theorem or property I have to use?) Thanks Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$ I have no idea where to even start. WolframAlpha cant compute it either.  $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$  I think it can be done with series, but I am not sure, can someone help a little so I can get a start??  Thanks! Does the Symmetric difference operator define a group on the powerset of a set? $G$ is the set of all subsets of a set $A$, under the operation of $\triangle\;$: Symmetric Difference of sets.  $A$ has at least two different elements.  I need to check if this is a group, and if it does to show if the group is abelian and/or finite.  Associativity - easy from Symmetric Difference. Identity element - empty group. Inverse element - each element is inverse to itself. Am I right?  abelian? finite? why does loop-cut-and-slide ignore some faces but not others? How can one obtain an adamantine weapon? Maybe I'm searching for the wrong thing, but I can't see how I can do this.  I have two servers:   Server A (has SQL Server 2012 installed) Server B (has SQL Server 2014 installed)   Server A provides an API via IIS. The code both inserts and gets data from the database depending on the API call.  Server B does exactly the same for its local database.  I would like to have it so that the two databases are consistent copies of each other. I do believe this is what HA accomplishes.   So a load balancer, for example, might direct API calls to A sometimes, and B sometimes, but the data needs to be consistent either way.  I found some documentation on HA for SQL Server 2012, but not on how to do this for a mixed environment of 2012 and 2014.  Any help or pointers to documentation would be great! Thanks.