Define $a_n$ is real sequence which satisfies   $$a_1=1, \quad a_{n+1}=\sin(a_n)$$  Find  $$\lim_{n\to\infty} \sqrt{n}a_n$$  I just know $$\lim_{n\to\infty} a_n = 0$$  but I don't know what should I do.  I'm interested in two solutions (using Stolz-cesaro, without using stolz-cesaro).
Consider the sequence defined as  $x_1 = 1$  $x_{n+1} = \sin x_n$  I think I was able to show that the sequence $\sqrt{n} x_{n}$ converges to $\sqrt{3}$ by a tedious elementary method which I wasn't too happy about.  (I think I did this by showing that $\sqrt{\frac{3}{n+1}} &lt; x_{n} &lt; \sqrt{\frac{3}{n}}$, don't remember exactly)  This looks like it should be a standard problem.   Does anyone know a simple (and preferably elementary) proof for the fact that the sequence $\sqrt{n}x_{n}$ converges to $\sqrt{3}$?
Where should looking-for-resources questions go? Escaping ampersand in URL Linux - Running The Same Command on Many Machines at Once I have about a dozen Linux boxes that I occasionally need to run the same command(s) on. Is there an easier way (or automated way) to do this asides from logging on to each machine and running the command, one at a time? It's not the same commands all the time and it's not at a preset time so it's not something ideal for tools like cron. How to pad zeroes to a string? What is a Pythonic way to pad a numeric string with zeroes to the left, i.e. so the numeric string has a specific length? I need to add a new field in the @book or @InProceedings entries: this field is "tome", for I have for instance the following title:  Francesco Bertolini, Società di trasmissione orale: mito e folclore, in Lo spazio letterario della Grecia antica, a cura di G. Cambiano, L. Canfora e D. Lanza, vol. I, tomo I, Salerno, Roma 1992, pp. 47--75.   So the following code is not enough:  @InProceedings{ bertolini:1992,   author                   = "Francesco Bertolini",   title                    = "Società di trasmissione orale: mito e folclore",   booktitle                = "Lo spazio letterario della Grecia antica",   editor                   = "G. Cambiano and L. Canfora and D. Lanza",   publisher                = "Salerno",   location                 = "Roma",   year                     = "1992",   volume                   = "1,   pages                    = "47--75", }   Using othes codes, I was able only to produce the "tome" field, but with a general trouble about the other fields:  \DeclareSourcemap{   \maps[datatype=bibtex]{     \map{       \step[fieldsource=tome,fieldtarget=usera]     }   } } \DeclareFieldFormat{usera}{tomo\space{#1}}   \DeclareBibliographyDriver{book}{%   \usebibmacro{author/editor}%   \setunit{\labelnamepunct}\newblock   \usebibmacro{title}%   \newunit\newblock   \usebibmacro{journal+issuetitle}%   \setunit{\bibpagespunct}%   \printfield{pages}%   \newunit\newblock   \printfield{usera}%   \usebibmacro{finentry}} \renewcommand*{\finentrypunct}{}   (useful for articles, but not for my case). By real, I'm not able to understand  which bibmascros I have to invocate...  I tried to write a biblatex-tome.dbx:  \ProvidesFile{biblatex-tome.dbx}   \DeclareDatamodelFields[type=field,datatype=literal,skipout=false]{tome}   \DeclareDatamodelEntryfields{tome}   Inside the main .tex file I put just this code:  \DeclareFieldFormat{tome}{\mkbibacro{tome}\addcolon\space #1} \newcommand*{\patchmoretome}[1]{   \xpatchbibdriver{#1}%     {\printfield{tome}}     {\printfield{tome}%           \newunit\newblock      \printfield{tome}}%     {\typeout{patching #1 to include more tome succeded}}%     {\typeout{patching #1 to include more tome failed}}% } % patch all drivers \patchmoretome{article} \patchmoretome{book} \patchmoretome{collection} \patchmoretome{inbook} \patchmoretome{incollection} \patchmoretome{inproceedings} \patchmoretome{manual} \patchmoretome{periodical} \patchmoretome{proceedings} \patchmoretome{report} \patchmoretome{thesis}   but it doesn't print the field in bibliographic entry. So I'm very confused about a (re)writing of driver Feed randomly gone on SE Android app SE Android App Version: 1.0.84  There is a display issue in the Feed section. The Earlier today, the below sections and the post title are not visible.  There is no exact steps to reproduce. It doesn't happen when the app is initializing. But if you open the Stack Exchange app, navigate to some other apps, then come back after few minutes, I can see this issue (but not all the time).  When the issue is there, if I manually click the Feed option in the navigation sidebar, the display issue is fixed.  I have recorded the issue: Why does Java have an "unreachable statement" compiler error? Projecting sp objects in R I have a number of shapefiles in different CRSs (mostly WGS84 lat/lon) that I'd like to transform into a common projection (likely Albers Equal Area Conic, but I may ask for help on choosing in another question once my problem gets better-defined).  I spent a few months doing spatial stats stuff in R, but it was 5 years ago.  For the life of me, I cannot remember how to transform an sp object (e.g. SpatialPolygonsDataFrame) from one projection to another.  Example code:  P4S.latlon &lt;- CRS("+proj=longlat +datum=WGS84") hrr.shp &lt;- readShapePoly("HRR_Bdry"), verbose=TRUE, proj4string=P4S.latlon)  # Shapefile available at  #   http://www.dartmouthatlas.org/downloads/geography/hrr_bdry.zip  #   but you must rename all the filenames to have the same  #   capitalization for it to work in R   Now I have a SpatialPolygonsDataFrame with appropriate projection information, but I'd like to transform it to the desired projection.  I recall there being a somewhat unintuitively-named function for this, but I can't remember what it is.  Note that I do not want just to change the CRS but to change the coordinates to match ("reproject", "transform", etc.).  Edit  Excluding AK/HI which are annoyingly placed in Mexico for this shapefile:  library(taRifx.geo) hrr.shp &lt;-    subset(hrr.shp, !(grepl( "AK-" , hrr.shp@data$HRRCITY ) |                                      grepl( "HI-" , hrr.shp@data$HRRCITY )) ) proj4string(hrr.shp) &lt;- P4S.latlon Clearing Flag Bits By Writing 1 Proof of upper-tail inequality for standard normal distribution $X \sim \mathcal{N}(0,1)$, then to show that for $x &gt; 0$, $$ \mathbb{P}(X&gt;x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \&gt;. $$ How to Disavow Bad Backlinks Automatically? Query works on Oracle 11g but fails on Oracle 8i Find splitting field of $x^5-4x+2$ over $\mathbb{Q}$ I know how to get the splitting field from the roots but I just can't come up with any factorisation that gives me roots. Any suggestions? It would be nice to allow users to mark a change in an answer (or question) as minor (e.g. for a simple spelling or grammar correction). A change marked as minor wouldn't push the question on top of the list of questions on the home page or in the feeds. It wouldn't trigger any notification. This could be implemented with a "Minor change" checkbox when editing an entry (a la Confluence). What do you think? I'm going to encrypt the characters Zhu, and decrypt them using RSA. I'm using the public key $\{e, n\}$ and private key $\{d, n\}$. The values of $e$, $d$ and $p$ I get from my textbook: $e = 17$, $d = 53$, $n = 77$.  It works fine when the number is smaller than 76, but in my example below, it doesn't work:  z = 90  h = 104  u = 107  e = 17 d = 53 n = 77  print z,h,u  c_z = z ** e % n c_h = h ** e % n c_u = u ** e % n  print c_z, c_h, c_u  p_z = c_z ** d % n p_h = c_h ** d % n p_u = c_u ** d % n  print p_z, p_h, p_u   The outputs are:  90 104 107 62 69 46 13 27 30     Do you have any idea what's going wrong? If I have the following code:  MyClass pClass = new MyClass(); pClass.MyEvent += MyFunction; pClass = null;   Will pClass be garbage collected? Or will it hang around still firing its events whenever they occur? Will I need to do the following in order to allow garbage collection?  MyClass pClass = new MyClass(); pClass.MyEvent += MyFunction; pClass.MyEvent -= MyFunction; pClass = null; Short Science Fiction story, told in a series of letters This was a short story told in a series of letters.  The protagonist starts by writing a letter to a utility company about an error in his bill.  By the last letter he is on death row trying to get his sentence commuted.  Each letter is slightly misunderstood.  I think it's from the late 60's and may have been in anthology like "Orbit". Recover data from hard drive with partitions (but not most data) overwritten I have a 500GB hard drive I've been keeping around to recover data from that I removed from a failing NAS drive that got sort of... erratic at the end. I finally got rid of the NAS when during a firmware update it removed the partition table.  Fast forward to a week ago, when I was building a new PC, and a mixup resulted in me placing the hard drive in question in the new PC and installing Windows XP on the first 100GB. I'm presuming any data on that first 100GB is now gone, but for the rest of it, is there any way I can recover it at home, as professional data recovery is currently too expensive?  I have a blank 1TB HDD if I can store any images of that hard drive on. The problem was definitely with the NAS and not the hard drive, as the hard drive had a successful install of Windows when mistakenly place in the new PC, and there were capacitors in the NAS's circuitry clearly broken.  The data I want to recover (in order of priority) is:   High: Some jpgs of family photos. Medium: Some RAW files. (There are also jpg versions of all of these) Low: Some mp3s, avis and ISOs, I can re-rip most of these if need be, but it'd be handy not to have to.    (I don't need a backup lecture, and if you can hold it in from nagging Jeff Atwood for it, you can hold it in from nagging me for it)  In short:   The partition tables are gone and overwritten. The data is not overwritten, except for an amount equal to the size of a Windows XP SP3 installation. I just bought a Ryanair plane ticket except that I did not see the QR code on the boarding pass, is that normal? the message is &quot;&quot;. thank you. Response.Redirect to new window I want to do a Response.Redirect("MyPage.aspx") but have it open in a new browser window. I've done this before without using the JavaScript register script method. I just can't remember how?