Find the limit of $a_{n + 1} = \frac{{1 + a_n + a_n b_n }}{{b_n }} $
Calculate the limit of two interrelated sequences?
How to interpret these acf and pacf plots Following are acf and pacf plots of a monthly data series. The second plot is acf with ci.type='ma':    The persistence of high values in acf plot probably represent a long term positive trend. The question is if this represent seasonal variation?  I tried to see different sites on this topic but I am not sure if these plots show seasonality.           Edit: following is the graph for lag upto 60:     Following are plots of diff(my_series):     And upto lag 60:     Edit: This data is from:  Here the contributors did not consider acf and pacf plot of original or differenced series worth mentioning (so it must not be important). Only acf/pacf plots of residuals was referred to in a couple of places. Cardinality of quotient ring $\mathbb{Z_6}[X]/(2X+4)$ What is the benefit/downside to using a switch statement vs. an if/else in C#.  I can't imagine there being that big of a difference, other than maybe the look of your code. Is there any reason why the resulting IL or associated runtime performance would be radically different? Related: I'm trying to connect 32 I²C "root of trust"  devices.    I'm thinking about a good way that does that, I mean what kind of communication should I use? A digital multiplexer that selects a device? but based on what? switching on the VDD source? but that needs reinitialization of the device and that takes time. i have a simple sort of a database file which consists only of entries in the following format     variable=value   i want to create a simple vim syntax highlighting for it and set it for specific file extension  for instance, variable part could be light blue, and value part light red  i googled it and came across things such as $vimruntime\syntax\, syntax set=, syntax match, and hi keywords, but couldn't set it up myself eventually  so i want a very simple vim code snippet that would realize it by matching the left and right hand sides and coloring them separately How to check if an object is nullable? How do I check if a given object is nullable in other words how to implement the following method... bool IsNullableValueType(object o) {     ... }  I am looking for nullable value types. I didn't have reference types in mind. //Note: This is just a sample. The code has been simplified  //to fit in a post.  public class BoolContainer {     bool? myBool = true; }  var bc = new BoolContainer();  const BindingFlags bindingFlags = BindingFlags.Public                         | BindingFlags.NonPublic                         | BindingFlags.Instance                         ;   object obj; object o = (object)bc;  foreach (var fieldInfo in o.GetType().GetFields(bindingFlags)) {     obj = (object)fieldInfo.GetValue(o); }  obj now refers to an object of type bool (System.Boolean) with value equal to true. What I really wanted was an object of type Nullable&lt;bool&gt; So now as a work around I decided to check if o is nullable and create a nullable wrapper around obj. Hyperref "enquotes" metadata Existence of area-preserving diffeomorphism between two sets in $\mathbb{R}^2$ Suppose that we have open sets $A, B\subset \mathbb{R}^2$ such that $A$ and $B$ are simply connected and have smooth boundaries. Furthermore, let us assume that $m(A) = m(B) &gt; 0$, where $m$ is the Lebesgue measure on $\mathbb{R}^2$. My question is as follows: when does there exist an area-preserving diffeomorphism between $A$ and $B$? In this context, an area-preserving diffeomorphism between $A$ and $B$ is a is a $C^{\infty}$-diffeomorphism $\phi : A\to B$ such that $m(S) = m(\phi(S))$ for all measurable $S\subseteq A$. If an area-preserving diffeomorphism doesn't always exist between sets $A$ and $B$ as above, are there any extra conditions we can impose on $A$ and $B$ such that there does exist an area-preserving diffeomorphism?  My thoughts thus far are that we always have a $C^{\infty}$-diffeomorphism between $A$ and $B$: this is a consequence of the . It's also obvious to me that if we can deform $A$ linearly to get $B$, then that deformation will be area-preserving. Furthermore, it seems that aside from this case, there's no reason why an area-preserving diffeomorphism should exist. However, I'm having a hard time proving that there necessarily isn't an area-preserving diffeomorphism between $A$ and $B$ if $B$ is not a linear deformation of $A$. Does anybody have any ideas? Points at integer distance How many points can one can place in $\mathbb{R}^n$, with the requirement that no $n+1$ points lie in the same $\mathbb{R}^{n-1}$-plane, and the euclidean distance between every two points is an integer? How to select top 10 records from each category This is a tricky one and I've always relied on techniques, such as permission-based emails (i.e. only sending to people you have permission to send to) and not using blatantly spamish terminology.  Of late, some of the emails I send out programmatically have started being shuffled into people's spam folder automatically and I'm wondering what I can do about it.  This is despite the fact that these particular emails are not ones that humans would mark as spam, specifically, they are emails that contain license keys that people have paid good money for, so I don't think they're going to consider them spam  I figure this is a big topic in which I am essentially an ignorant simpleton. How can I use sed to replace a multi-line string? I've noticed that, if I add \n to a pattern for substituting using sed, it does not match. Example:  $ cat &gt; alpha.txt This is a test Please do not be alarmed  $ sed -i'.original' 's/a test\nPlease do not/not a test\nBe/' alpha.txt  $ diff alpha.txt{,.original}  $ # No differences printed out   How can I get this to work? Audio is working, but the speaker test doesn't work I've installed Ubuntu 12.04 LTS from a minimal cd on a netbook (Asus 1001 PXD). I've installed the ubuntu-desktop package using the --no-install-recommends option.  Everything works fine, except the "sound test" for headphones or analog speakers.  Clicking on the "test" buttons (front left and front right) I don't hear any sound. Despite this, the audio is working properly.  I've checked the audio levels with alsamixer; I have also checked that the test sounds actually exist in /usr/share/sounds/alsa.  I tried an installation of Ubuntu 12.04 LTS made with the desktop-cd, and in this case the speaker test works properly.  I suppose, therefore, that the problem could depend on the lack of a package, but I have not identified which one. OS X Yosemite brought support for recording iPhone/iPad screen natively without having to using AirPlay applications like AirServer or Reflector.   How do I record the iPhone screen on macOS ? Is password confirmation necessary on a Change Password dialog? Questions about    setting environment variables the PATH   are very common here, and in most cases the answers are very similar to each other. In the future it would be nice to have a good Q/A for this.  So the question is: What are environment variables, like the executable PATH, and how can I change and use them on major operating systems?  A good answer would include a simple explanation of what environment variables and especially PATH mean to the OS, as well as simple guidelines on how to set and read them accordingly. Why is there no absolute maximum temperature? HTML element does not fade in using jquery How do I find out which repository a package comes from? Is there a method or command which can tell from which repository a package is coming from? I heard the following comment of Mr. Marco Rubio about Mr. Donald Trump on CBS’s “Face the Nation” in a clip in February 29th AP Radio News:  “There is never gonna be a time where the Republican Party rallies around and says you have to get out or anyone has to get out for purposes of rallying around Donald Trump. Donald Trump is not Republican Conservative. Donald Trump is trying to pull off the biggest scam in American political history.”  I thought I misheard “There is never gonna be a time where –“ for “There is never gonna be a time when –“ at first, because I was in understanding that time should be trailed by “when” as a relative noun or adverb. An English Japanese dictionary at hand (Readers English Japanese Dictionary published by Kenkyusha) provides “where” singly as a relative to refer the ‘place.’ However, my lookup of the following two sources endorsed that I heard it right:  Here is what is never going to happen in this race. There's never going to be a time where the Republican Party rallies around and says you have to get out or anyone has to get out for purposes of rallying around Donald..  “Here is what is never going to happen,” Rubio continued. “There's never going to be a time where the Republican Party rallies around and says you have to get out or anyone has to ...   Is it common, at least not uncommon to use “where” after “time” as a relative?