I got stuck in solving a question of power series in complex analysis in Conway's exercise. Which is as follows $:$  Find the radius of convergence of the power series $:$ $$\sum_{n=1}^{\infty} \frac {(-1)^n} {n} z^{n(n+1)}.$$ What happens at the boundary of the disk of convergence?  I found that the radius of convergence to be $1$ and hence the disk of convergence is $|z|=1$. Now I found at $z=1,-1$ the series is convergent by Leibnitz test. Now how can I check the point of convergence (if any) in the disk of convergence other than $1$ and $-1$. I tried with Dirchilet's test. But I couldn't actually figure out how is $$\sum_{k=1}^{n} z^{k(k+1)}$$ look like? Is it bounded for arbitrary $n$ for all $z$ on $\partial B(0;1)$ except at $1$ and $-1$? Please give me some suggestions. Thank you in advance.
I already found the radius of convergence of the power series $:$ $$\sum_{n=1}^{\infty} \frac {(-1)^n} {n} z^{n(n+1)}.$$ What happens at the boundary of disk of convergence?  I found that the radius of convergence to be $1$ and hence the disk of convergence is $|z|&lt;1$. Now I found at $z=1$ the series converges. Now how can I proceed? Please help me. Thank you in advance.
How can I read input from the console using the Scanner class in Java? Where should I put my bash scripts How to install Flash player on Ubuntu? Given a large (~ 1 million) sample of unevenly distributed points - is it possible to generate irregular grid (in size, but could also be irregular in shape if that is possible?) that will contain specified minimum amount of n points?   It is of less importance for me if genereted 'cells' of such grid contain exactly n number of points or at least n points.  I'm aware of solutions like  in ArcGIS or  in QGIS/mmgis however they will all create regular grids which will result in an output with empty cells (smaller problem - I could simply discard them) or cells with count of points less than n (bigger problem since I'd need a solution to aggregate those cells, probably using some tools from ?).  I've been googling around to no avail and am open to both commercial (ArcGIS &amp; extensions) or free (Python, PostGIS, R) solutions. I want to connect my laptop (Inspiron 15 5000 Series) to a 4k display. When I read online about this it seems that HDMI 2.0 is a necessaity to handle the large resolutions / refresh rates involved to get full benefits of a 4k display.  Use case: I'm not going to use this to play games. Most of my use-case is editing documents, emails, spreadsheets and some engineering drawings and video (watching not editing).   The goal behind going 4k is to boost productivity by having more things open at once to read side by side.   I looked into the Dell specs and they only mention an HDMI without going into the details of what type or whether it will support HDMI 2.0 / 4K TV.     How do I tell whether my Dell Inspiron Laptop (i7 7500 U processor) supports this?  I looked under the Display Adapters and can see that I have an "Intel HD Graphics 620"  listed.   Will this suffice to tell me if this display adapter will support HDMI 2.0 / 4k display?   Or should I be checking out some other settings?  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Number of ways a natural number can be written as sum of smaller natural number It is easy to realize that given a natural number N the number of doublets that sum N are $\frac{N+(-1)(N \pmod 2)}{2}$ , so I thought I could reach some recursive formula in the sense that found the number of doublets I could find the number of triplets and so on ...,  example: N=3 the only doublet is 2+1=3 -not said yet but 2+1, and 1+2 count as one- then I could count the number of way the number 2 can be expressed as the indicated sum and got the total number of ways 3 can be written as a sum. But this seems not so efficient, so I was wondering if there is other way to attack the problem and if there is some reference to this problem such as if it is well known its used, once I read that this have a chaotic behavior, and also read It was used in probability but don't remember where I got that information. So if you know something I would be grateful to be notice, thanks in advance. Infinity, is clearly not . But there is some dissent on whether or not zero is finite. I have seen authors use "finite" to indicate the value of $0$ is excluded as well as infinity.  Is there any rule about this? Does the given matrix have a square root? Given the following matrix, I was asked to tell whether the matrix is the square of some matrix or not.  $$ \begin{pmatrix} 1 &amp; 0 &amp; 0\\ 0 &amp; -1 &amp; 0\\ 0 &amp; 0 &amp; -1 \end{pmatrix} $$  Since this matrix is diagonal, a square root is  $$ \begin{pmatrix} 1 &amp; 0 &amp; 0\\ 0 &amp; i &amp; 0\\ 0 &amp; 0 &amp; i \end{pmatrix} $$ Am I right? call function declared below Is it possible to call a function which is declared below in bash?  Example  if [ "$input" = "yes" ]; then     YES_FUNCTION elif [ "$input" = "no" ]; then     NO_FUNCTION else     exit 0; fi  YES_FUNCTION() {   .....   ..... }  NO_FUNCTION() {   .....   ..... } using gb4e keeping interlinear glosses together on the same page I am using gb4e for several hundred interlinear glosses in my dissertation. Several of the examples span more than one page, that is, the vernacular and gloss will be on one page, while the free translation appears at the top of the next page.   How can I keep all three elements together on the same page? I often have subexamples (a, b, c) that don't need to stay together. That is, 1a can be on one page, and 1b on the next, as long as vernacular-gloss-free translation for each subpart stay together.  In the MWE below, the free translation is at the top of page 2, while the vernacular and gloss are at the bottom of page 1.  \documentclass[11pt,letterpaper]{article} \usepackage{lipsum} \usepackage{gb4e} \noautomath \begin{document} \lipsum[1-4] \begin{exe}     \ex \gll wuz x̃ũch it-ak xohish-em\\              1\textsc{sg.prn} bread eat-\textsc{inf} want.\textsc{prs}-1\textsc{sg}\\         \glt `I want to eat bread.'  \end{exe} \end{document}   A solution was proposed to a similar question (), but it requires me to manually look for what could be dozens of orphans throughout my dissertation (and doesn't work when applied to my example, since it still strands the FT). I would prefer a more elegant solution that takes care of the problem automatically. Are there rules about using "that" to join two clauses? What are point releases in LTS versions? There are so called "point releases" for LTS versions of Ubuntu. 10.04.3 for instance.  What are those "point releases" releases? How's This for a Simple HTTP Callout Retry Framework? Most of what I have read about retry operations is that they are vaguely complicated and indepth. I have written the code below which seems to accomplish this without too much confusion - I'm wondering if there are any pitfalls I am not seeing here.   The class uses a top-level schedulable class, which calls a queueable class to make the callout, and if the callout is unsuccessful, it just schedules the top level class again five minutes later. After testing it appears this works indefinitely.   A simple string token, assigned at the first invocation, is also passed between the classes, and I have tested and found that the token does persist, even when runs are scheduled for a later time.   Thanks for helpful thoughts!  public class SelfServe implements Schedulable {      public String Token = null;      public void execute(SchedulableContext ctx) {                  RetrieveMachineDetails Job = new RetrieveMachineDetails();             Job.Token = Token;             System.enqueueJob(Job);      }           public class RetrieveMachineDetails implements Queueable, Database.AllowsCallouts {          public String Token = null;          public void execute(QueueableContext context){                                            Http http = new Http();             HttpRequest request = new HttpRequest();                             request.setEndpoint('https://app.endpoint.com/api');                     request.setMethod('POST');                     request.setHeader('Content-Type', 'application/json;charset=UTF-8');                                      request.setBody('{"query":  "$message_=\'METRIC\' &amp;&amp; $tier=\'customer\' "}');                                                                                                                                                                                    HttpResponse response = http.send(request);                            if (response.getStatusCode() == 200) {                 //do stuff                  }              else { //retry                  DateTime Dt = system.now().addMinutes(5);                                String day = string.valueOf(Dt.day());                                        String month = string.valueOf(Dt.month());                                          String hour = string.valueOf(Dt.hour());                                       String minute = string.valueOf(Dt.minute());                                                       String year = string.valueOf(Dt.year());                                   String ScheduledTime = '0 ' + minute + ' ' + hour + ' ' + day + ' ' + month + ' ?' + ' ' + year;                  SelfServe Job = new SelfServe();                     Job.Token = Token;                 System.schedule('SelfServe Job'+Dt, ScheduledTime, Job);              }         }        }  } "Where" exactly are complex numbers used "in the real world"? I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be used/applied and hence am curious.  So what practical applications of complex numbers exist and what are the ways in which complex  transformation helps address the problem that wasn't immediately addressable?  Way back in undergrad when I asked my professor this he mentioned that "the folks in mechanical and aerospace engineering use it a lot" but for what? (Don't other domains use it too?). I'm well aware of its use in Fourier analysis but that's the farthest I got to a 'real world application'. I'm sure that's not it.  PS: I'm not looking for the ability to make one problem easier to solve, but a bigger picture where the result of the complex analysis is used for something meaningful in the real world. A naive analogy is deciding the height of tower based on trigonometry. That's going from paper to the real world. Similarly, what is it that is analyzed in the complex world and the result is used in the real world without imaginaries clouding the problem?  The question:  is nice but covers a more mathematical perspective on interim results that make solving a problem easier. It covers different ground IMHO. How do I execute the following shell command using the Python  module?  echo "input data" | awk -f script.awk | sort &gt; outfile.txt   The input data will come from a string, so I don't actually need echo.  I've got this far, can anyone explain how I get it to pipe through sort too?  p_awk = subprocess.Popen(["awk","-f","script.awk"],                           stdin=subprocess.PIPE,                           stdout=file("outfile.txt", "w")) p_awk.communicate( "input data" )   UPDATE:  Note that while the accepted answer below doesn't actually answer the question as asked, I believe S.Lott is right and it's better to avoid having to solve that problem in the first place! Objects disappear when looking through camera When I go to camera perspective, there are parts of my object that I can't see. I tried adding and deleting my camera a lot of times and positioning it so that the objects I would like to see on my camera view will be seen but to no avail, I still get the same problem.    What could be wrong with my camera set-up? Gandalf (as with all the Istari) did not have a corporeal form before coming to Middle-earth and was essentially ageless. We also know that Gandalf did age, albeit very slowly, while on Middle Earth. By the end of the Third Age (before battling Durin's Bane), he had the appearance of an old human male (non-Númenórean, etc) in his 70's or 80's. This leads me to wonder:  Is there any indication in Tolkien's writings that describes Gandalf's appearance (in human-looking years) when he arrived in Middle Earth? In other words, was Gandalf the Grey originally "Gandalf the Tall Dark &amp; Handsome in his 20's"?  I'm not talking about his actual age, but rather his appearance in human years. I already have the pink and red leather gloves, the armor, the crown, and the monkey's staff (and they're all enchanted). Regardless of how many health potions I use, I always die. Help!