Sum of infinite series by hand
Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$
Significance contradiction in linear regression: significant t-test for a coefficient vs non-significant overall F-statistic Assume $R=(\mathcal{U}_n)_n$ is a sequence of distinct ultrafilters on some set $X$. Since every Hausdorff space has an infinite discrete subspace, there is a subsequence $R=(\mathcal{V}_n)_n$ of $R=(\mathcal{U}_n)_n$ such that $\{\mathcal{V}_n|n \in \mathbb{N} \}$ is a discrete subspace of $\beta X$. In particular, there is a sequence $(A_n)_n$ of sets with $A_n \in \mathcal{V}_m$ if and only if $m =n$. If we let $B_n=A_n - (A_0 \cup...\cup A_{n−1})$, then the sequence $(B_n)_n$ is pairwise disjoint and $B_n \in \mathcal{V}_m$  iff $m=n$. If we let $B =\cup_n B_{2n}$, then $B \in \mathcal{V}_m$ if and only if $m$ is even. In other words, if $B =\{ \mathcal{V} \in βX|B \in \mathcal{V}\}$, then $B$ is a clopen set with $\mathcal{V}_m \in B$ iff $m$ is even. Therefore, we conclude that the sequence $(\mathcal{V}_m)_m$ cannot converge to any point, so the sequence $(\mathcal{U}_n)_n$ cannot converge to any point either.     I would like to know that:      (1) Why "$B_n \in \mathcal{V}_m$  iff $m=n$"? And " $B =\cup_n B_{2n}$, then $B \in \mathcal{V}_m$ if and only if $m$ is even "?      (2)$B$ is a clopen set with $\mathcal{V}_m \in B$ iff $m$ is even How to use st_buffer? I would like to calculate a buffer of 5km around country polygons. However, I somehow always get a warning and an error. Here is a reproducible example code.   library(sp) library(sf) library(tmap)  library(rnaturalearth) countries_sp &lt;- ne_countries(scale = 10)  countries_sp &lt;- countries_sp[(countries_sp@data$geounit %in% "France"), ]  proj4string(countries_sp) &lt;- CRS("+init=epsg:4326")   countries_for_buff &lt;- st_as_sf(countries_sp, crs ="+proj=laea +lat_0=52 +lon_0=10 +x_0=4321000 +y_0=3210000 +ellps=GRS80 +units=m +no_defs ")  countries_buff_5km &lt;- st_buffer( countries_for_buff, 5000)   here the warning:  dist is assumed to be in decimal degrees (arc_degrees). Warning message: In st_buffer.sfc(st_geometry(x), dist, nQuadSegs, endCapStyle = endCapStyle,  : st_buffer does not correctly buffer longitude/latitude data   Then   tm_shape(countries_sp) +   tm_polygons() + tm_shape(countries_buff_5km) +   tm_polygons()   gives the error   Error in if (all(tiny) &amp;&amp; show.messages) warning("all relative area sizes are below thres.poly",  :     missing value where TRUE/FALSE needed   What am I doing wrong? I guess there is something about the projection that does not quite work out the way it should. Modern ice-age/plans for weapon hidden in genome How to get the list of IP addresses in a LAN? How to draw a crisp, opaque hairline in JavaFX 2.2? Update: Universal login is in place and considered &quot;done&quot;. If you see anything weird, please submit new bug reports.   As y'all know, our current flavor of &quot;global authentication&quot; leaves a few things to be desired. It's flaky, requires a page refresh, etc. etc. Wouldn't it be nice if you could just sign in once and be automatically logged in across the network?  We are ready to roll out Stage 1 of Project &quot;Make Authentication Suck Less&quot;. It is live as of a few minutes ago. What this means right now:  When you log into any Q&amp;A site on the stackexchange.com domain, you will be automatically logged into all other Q&amp;A sites on the stackexchange.com domain + stackexchange.com itself.   is excluded from this.   , , and other Q&amp;A sites with their own domains will still use the old flow.  We are going to recognize you in some ways on sites where you don't have a profile. This is still a work in progress, but most notably you will see &quot;join this community&quot; in the top bar instead of &quot;sign up&quot; and &quot;log in&quot;.   But I use multiple accounts! What do? We aren't intending to break the ability to run multiple accounts. The same guidelines as before still apply — we don't officially support that, you might end up with an accidental automatic merge if you're not careful with your credentials, any interaction between the accounts is right out, and so on. If you're already using different browsers or different browser profiles for your accounts, nothing will change for you. If you are somehow logged into multiple accounts in the same browser... I'm sorry to say, but that's a bug even for our current global auth setup. You will now only be able to log into one at a time, and you'll need to log out when you want to switch. (Or start using different browsers.) What's next:  Remember  we tested a few months ago? It's coming back. Stage 2: bring other domains into the fold (this is also live now) Stage 3: make cross-domain experience smoother and support browsers that reject third-party cookies in all conditions (looking at you, Safari) - this is not currently feasible. Safari users and users who reject third-party cookies will need to sign in separately to every second-level domain (e.g. stackexchange.com, stackoverflow.com, etc).  If you see any issues… Good luck, because you probably can't log in to report them here. =) But seriously, let us know via the contact form linked at the bottom of any page. If you are able to log in, report issues here on meta as normal. So, in conclusion… Compare using Thread.Sleep and Timer for delayed execution I have a method which should be delayed running for a specified amount of time.  Should I use  Thread thread = new Thread(() =&gt; {     Thread.Sleep(millisecond);     action(); }); thread.IsBackground = true; thread.Start();   Or  Timer timer = new Timer(o =&gt; action(), null, millisecond, -1);   I had read some  about using Thread.Sleep is bad design. But I don't really understand why.  But for using Timer, Timer has dispose method. Since the execution is delayed, I don't know how to dispose Timer. Do you have any suggestions?  Or if you have alternative codes for delayed execution are also appreciate. mysql_insert_id alternative for postgresql is there an alternative for mysql_insert_id() php function for PostgreSQL? Most of the frameworks are solving the problem partially by finding the current value of the sequence used in the ID. However, there are times that the primary key is not a serial column.... bash aliases equivalent for powershell? By default my Windows PowerShell starts in C:\Users\Santosh, my XAMPP installation is in D:\ so the htdocs folder is located at D:\xampp\htdocs. If I have to edit something in htdocs folder then I have to type full cd D:\xampp\htdocs\ (autocompletion is not so kind) then edit that file.  If this PowerShell were a Bash I would do this in .bash_aliases file:  alias htdocs='cd D:\xampp\htdocs'  Is it possible to maintain Bash aliases like file and alias any command in PowerShell? On iOS, while watching a video in YouTube app, when I go to the Home Screen, the video stops playing. I found no setting to enable it.  I know it is available for Android, but is there a way to do so in iOS? Is there any setting, or a 3rd party app that I can use? Writing to NTFS drive in Linux I've got an external USB drive that is formatted with NTFS mounted on ~/tmp.  I can read it just fine, but cannot write to it, even as root (temp is a folder on the USB drive):  [root@Plugbox temp]# touch test touch: cannot touch `test': Permission denied [root@Plugbox temp]#   However, if I do a cat /proc/mounts, it looks like I do have NTFS read/write support:  [root@Plugbox temp]# grep /dev/sdb1 /proc/mounts /dev/sdb1 /root/tmp ntfs rw,relatime,uid=0,gid=100,umask=02,nls=utf8,errors=continue,mft_zone_multiplier=1 0 0 [root@Plugbox temp]#   Finally, if I go to the folder above temp, it looks like I should have plenty of permissions to write to the folder:  [root@Plugbox tmp]# ls -al total 61500764 ... drwxrwxr-x 1 root users           0 Apr 19  2010 temp ... [root@Plugbox tmp]#   What am I missing? Can't redirect cut output Is the tag info page not available on mobile theme? I was surfing SO on my iPhone, and wanted to see the info (tag-wiki) page for a tag, but didn't find it. Clicking on the tag always leads to the list of question, and there is no other link to the tag wiki. Is it not available on the mobile site? Why did It not do this to The Losers? How do RAM and processor speed affect overall performance on Android? Since Android can be installed on wide range of devices, I'm wondering how important RAM and the CPU speed are to the performance of a device.  There are always tradeoffs when something is increased and another one decreased!  Is more RAM or a faster processor better for general browsing and less mulititasking?  What if I'm interested in gaming or doing lots of things at once?  Will more RAM make up for a slower processor or vice-versa? Could you help me finding some less colloquial equivalent to highlighted phrases of this speech Today I want to be able to give you some good tips on parallel parking. It's part of your road test. It's something you need to learn. For most new drivers it's real difficult, because they're not able to judge angles. And parallel parking is all about angles. I'll take you through a typical parking, I'll take you through how I teach it , and maybe that'll help you in passing your road test. The first thing you want to do is you want to pull over over and you want to stop alongside a parked car. Typically, an inspector will give you a command to pull over, stop next to the red or blue car. You're just going to pull over, put your blinker on to the right, and pull over even with that car.     taken from a series of videos teaching driving   Why did the man use  be able in his speech? What is the difference between be able and I would like to or I want ? I mean what the listener should make of  I want to be able to give you ...  . Why did not he say I want to give you... ?  When he says I'll take you through a typical parking , I get the meaning of take one through ,It means I am going to teach you , isn't it? But my real concern is what the second I'll take you through means, while saying I'll take you through how I teach it . Perhaps it doesn't mean I will teach you that how I teach it ! , does it? The man always says the first thing you want to do . Why does not he say the first thing you should do , the first thing you need to do ? What are the diffrences in meaning? Isn't it unnatural saying you want to do ? In the last sentence he says You're just going to pull over, is the use of going to just weakening would and will? I think here it means something else. I think it means you just want to or you just start Explicitly finding the sum of $\arctan(1/(n^2+n+1))$ This is from a GRE prep book, so I know the solution and process but I thought it was an interesting question: Explicitly evaluate $$\sum_{n=1}^{m}\arctan\left({\frac{1}{{n^2+n+1}}}\right).$$ How can i disable totally the prompts that appear while installing a Debian package, i've used all the options that i've found but there are some packages that are still prompting.  I'm using this command:  apt-get -y --allow-unauthenticated --force-yes -o DPkg::Options::="--force-overwrite" -o DPkg::Options::="--force-confdef" install x11-common   Why the x11-common package is still prompting? how can i get rid of these prompts?  Thanks in advance  --Victor  Edit: just to clarify, the prompts are not "yes/no" prompts, are open questions in a coloured screen (typical two color screen) but i want to set the default option of these questions My linear algebra professor proved that      Diagonalizable matrices with complex values are dense in set of $n \times n$ complex matrices.   He defined a metric (I believe) that was somehow related to the usual metric on $\mathbb{R}^{n^2}$.  Then somehow proved that diagonalizable matrices were dense because for any matrix $A$ if $\det(A - \lambda I) = 0$ on an open subset, then $\det(A - \lambda I)$ was the zero polynomial.  I Googled around a bit and found some stuff talking about the Zariski topology, and I am not sure this is what I want. Most of what I have found on this topology is much more general than what he was doing.  Does anyone know any resources that would explain how to go about proving this statement?  Edit: I found out how to prove this the way my professor did.  We define the norm of a matrix by $$ |A| = \max\{|Ax| : |x| = 1 \}.$$ Now we have a distance $d(A, B) &lt; \epsilon$. You can prove that if $(A - B)_{ij} &lt; \epsilon/n$, then $|A - B| &lt; \epsilon$.  Let $p(x)$ be the characteristic polynomial of $A$, an $n \times n$ matrix.  $$p(x) = \prod_1^n (x - x_i) = x^n + (-1)\sigma_1 x^{n - 1} + \cdots + (-1)^n\sigma_n$$ where $\sigma_1, \ldots, \sigma_n$ are the elementary symmetric polynomials of $x_1, \ldots, x_n$ which are the eigenvalues.  The discriminant of the characteristic polynomial is a symmetric polynomial, therefore it can be written in terms of the elementary symmetric polynomials, which in turn can be written in terms of the entries of the matrix.  But since the discriminant is a polynomial, it only has finitely many roots. Therefore for $\epsilon &gt; 0$, by changing the entries of the matrix less than $\epsilon / n$ we can find a new matrix $B$ such that $|B - A| &lt; \epsilon$ and the discriminant is not zero.   The discriminant being not zero means $B$ has distinct eigenvalues, thus has a basis of eigenvectors. Therefore, it is diagonalizable.  Thus the set of diagonalizable matrices is dense in the set of matrices with respect to that metric.