Prove that this specific sequence converges
Proof that a sequence is convergent
Are there guidelines on how much free RAM a phone should have - specifically Samsung Galaxy S? I have a Samsung Galaxy S (AT&amp;T Captivate) that is rooted.  I have various ways to tweak the amount of free RAM the OS leaves on the phone.  The default is 48 MB, but I was wondering if I could actually go lower without impacting phone performance.  Are there guidelines posted anywhere, or that can be deduced based on knowledge of the Android platform?  Or should I trust that the default settings are appropriate?  I understand that when it comes to free RAM, more isn't necessarily better, but I understand that there is still a threshold that I don't want to dip below.   I'm running a 2.3.6 Gingerbread based ROM.  For what it's worth, I'm running the Zeam launcher. Why are the solutions of polynomial equations so unconstrained over the quaternions? An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers.  But it may have an uncountable set of zeroes in the quaternions.  For example, $x^2+1$ has two zeroes in $\mathbb C$, but in $\mathbb H$, ${\bf i}\cos x + {\bf j}\sin x$ is a distinct zero of this polynomial for every $x$ in $[0, 2\pi)$, and obviously there are many other zeroes.  What is it about $\mathbb H$ that makes its behavior in this regard to be so different from the behavior of $\mathbb R$ and $\mathbb C$?  Is it simply because $\mathbb H$ is four-dimensional rather than two-dimensional? Are there any theorems that say when a ring will behave like $\mathbb H$ and when it will behave like $\mathbb C$?   Do all polynomials behave like this in $\mathbb H$? Or is this one unusual? Derivation of the formula for Ordinary Least Squares Linear Regression Same as title, want to kill alt-tab, etc. What's the difference between stochastic and random? How to get scroll position from GridView? Give $a, b. c$ be the lengths of a triangle and $a+ b+ c= 3$. Prove that: $$\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\geq  \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$$ My try  We have England MO inequality: $$\left ( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a} \right )^{2}\geq  \left ( a+ b+ c \right )\left ( \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c} \right )$$  and $$\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a} \geq  3=  a+ b+ c$$  But it cannot be used to prove the above inequality and how to use triangle inequality in this inequality. I need the help. Thanks! How to show every field is a Euclidean Domain. I'm having trouble proving this. This is what I have so far:  Let $F$ be a field. Let $v(x) \rightarrow 1$ for all $x$ not equal to $0$.  So if we let $x$ be in $F$ where $x$ not zero then we can write $x$ as: $x=qy+r$ for some $y$ in $F$. If $r$ not zero then $V(r)=1$.  Not sure what to do from here. I know eventually I'm supposed to get no remainder $(r=0)$ but I'm stuck at applying the definition and valuation map. Now that I'm looking into home automation, I'm thinking about automatically cutting off the fridge power from 3am ~ 5am. Given that modern refrigerators are well insulated, it should be able to withstand 2 hours of non powered cooling.  If I proceed with this project, will I see a big reduction in my monthly electrical bill? Is there software on Macintosh for adding GPS data to RAW files? Share menu hangs iOS app The Share menu in iOS 13 got a redesign, and the Stack Exchange iOS app is no longer compatible with it. I can still use it to copy the link, but when I try close the menu, it remains stuck. Opening another app and then returning automatically dismisses the menu, but then I can't use the back button anymore to return to the question list. (I'm still able to refresh the page, so it's a navigation issue.)    Ironically, I discovered this bug while trying to mark  as a duplicate; in the app, the quickest way to do that is to use the Share menu to copy the link to the original question to the clipboard...   App Version: 1.6.6.2 Device: iPhone 7 Plus OS Version: Version 13.1 (Build 17A844) How to make this curly 'R' (ℛ)? Which package has a similar looking fancy R or does anyone know how to make one? (Note: the line underneath the R is just the notebook paper... this is from a scanned set of notes). Chess Piece Combinations I came across a question yesterday about combinations, and I wanted to know what the correct answer was. The question states as follows:  There are 8 spaces that are alternately black and white. There is one king, one queen, two identical rooks, two identical bishops, and two identical knights. The king needs to be surrounded by the two rooks. Then, the two bishops can be put on any of the remaining spaces as long as one is on a black space and the other is on a white space. Finally, the queen and the knights can be placed anywhere in the remaining spaces in relation to the other pieces. How many possible arrangements are there considering these conditions? Why did the limit of power change from wizard/witch to wand? So in the first 5 or 6 books, it was very clear that the wand chooses the wizard and the power/level of the spells the wizard produces is based on the knowledge and skill of the wizard.  This is referenced by people constantly being telling Harry that he wasn't skilled enough to perform certain spells, or Bellatrix telling Harry that he needed to mean the Cruciatus curse for it to be effective.  However, in the 7th book, this entire idea is changed to the wand being the thing that's powerful and regardless of the wizard's/witch's skill or knowledge, he/she could still perform a very advanced spell if it's something the wand could handle.  Goyle accidentally creating a fire serpent is something that instantly comes to mind from the movies(I don't remember if that happened in the book).  So, my question is, what actually holds the power, the wizard/witch, or the wand? I am fairly new to Unity3D experience , but I have a very good knowledge of Java and Android development. I am really confused that if Java is at all needed for developing android applications? I read somewhere in Unity documentation that adding behaviour to objects in Unity requires use of scripts, and that unity only supports C# , .Net , and Boo scripts. Is there no use of Java at all? clerics have the ability  (PHB, p. 62):     When a creature within 5 feet of you that you can see hits you with an attack, you can use your reaction to cause the creature to make a Dexterity saving throw. The creature takes 2d8 lightning or thunder damage (your choice) on a failed saving throw, and half as much damage on a successful one.   It doesn't specify what the DC of the saving throw is.   I assume it uses the spell save DC, even though it technically isn't a spell?     Spell save DC = 8 + your proficiency bonus + your Wisdom modifier Does 'but for' mean:  'If we had X (but we didn't), Y would have been the consequence'?  Or can it also mean; because we had X, as a result Y happened?   Some different examples of but for:  (Case 1)     But for a cup coffee, it was to be a productive day of writing.   Unfortunately, I'd run out, and spent the day craving.    (Case 2)     But for the glass of wine I had dinner, the house work didn't get   done.   (Case 3)     But for the glass of wine I had for dinner, I would have got the   house work done   What is the correct usage of "but for"? How can I determine whether a 2D Point is within a Polygon? I'm trying to create a fast 2D point inside polygon algorithm, for use in hit-testing (e.g. Polygon.contains(p:Point)). Suggestions for effective techniques would be appreciated. Six simple tips to get Stack Overflow reputation fast From "6 Simple Tips to Get Stackoverflow Reputation Fast" at codexon.com:   Be the First to Answer. Even at the cost of quality. Use Downvotes and Comments Strategically Use obnoxious in-your-face formatting and lists. Be Aware of the 200 rep/day Limit Edit, But Don’t Edit Too Much Associate your other accounts   Courtesy of . Agree? Disagree? Walnuts? Cantaloupe? Imported 3D scan displays texture in editor, but renders without texture I am new to blender and I have a 3D scan of a stone tablet (which I am unfortunately not able to share, so I will try to be as detailed as possible) which was imported as OBJ and MTL files.  The editor displays the texture and mesh perfectly, but when I render the scene (pressing F12) the texture is not there.  I am trying to use blender to reduce the file size of the tablet for use in a three.js visualizer for a website.  I have tried exporting the file as a JSON file (as well as several other file formats to the same effect) for use in my three.js webapp and thought it was an issue with three.js, but after seeing that blender did not render the tablet with the texture I now believe it is an issue with my blender configuration. I am unable to find within blender or online any way to check what current resources I have displayed and what options I have with them.  I have not applied a mesh or texture to the object, but when I do so it renders with the generated texture applied in the default pink.  I am confused as to where the disconnect is between the editor and the renderer. I am still figuring out what is important and where to find all the relevant information, so any help on that regard would be very much appreciated.  Here is what blender looks like for me with the sensitive parts censored (can't be too careful these days):   I have looked at this question:  And there is no material overwriting the texture - I have not assigned a material to the tablet at all. EDIT: In regards to the duplicate answer, I looked at that one as well and the issue I have is that it doesn't have an actual texture file that I can just put on it.  The OBJ MTL files come with their own file that can't be applied directly to the model.  Blender read the OBJ MTL files and recreated the object inside blender algorithmically - and now I need to know how to get blender to tell me where that texture is and how I can get it to show up in the render.  The texture is not on my hard drive, but entirely within blender which is the issue I am having.