Convergent infinite sum
Sum with parameter convergence
Interpreting the end of Inception Sorry for the vague title--I don't want the title to be a spoiler.  I've seen the movie three times, and the last two times I looked specifically for clues to this question, but found none.  The final scene in  shows       Cobb's top spinning, as he goes off to be reunited with his children.  The top continues to spin, then the movie ends, without telling us if the top topples--and therefore revealing whether he's in the "real world" or still in a dream. Do we know if he was in a dream at the end of the movie?   I'm assuming the answer is that we simply cannot know.  But did I miss anything?  Were there clues that I missed? Showing $F[x,y]/(ax^2+by^2-1)$ is a Dedekind domain A quick search for currency regex brings up . The problem I have in choosing one is that regex is difficult to verify without testing all the edge cases. Does anyone have a regex for U.S. currency that has been thoroughly tested? My only requirement is that the matched string is U.S. currency and parses to System.Decimal:  [ws][sign][digits,]digits[.fractional-digits][ws]   Elements in square brackets ([ and ]) are optional.  The following table describes each element.   ELEMENT             DESCRIPTION ws                  Optional white space. sign                An optional sign. digits              A sequence of digits ranging from 0 to 9. ,                   A culture-specific thousands separator symbol. .                   A culture-specific decimal point symbol. fractional-digits   A sequence of digits ranging from 0 to 9. QGtkStyle was unable to detect the current GTK+ theme When I start any Qt app, I get this message in the terminal, and the Qt app doesn't use the same theme as my Gtk apps:  QGtkStyle was unable to detect the current GTK+ theme.   How do I make QGtkStyle detect my current GTK+ theme?  It seems that LXDE is using a settings daemon, and there is no gtkrc-2.0 file in my home folder. I'm not quite sure how this happened, but somehow a completely empty hierarchy of directories has ended up in my repository:  com/ com/companyname/ com/companyname/blah/ com/sun/ com/sun/java/ com/sun/java/jax_rpc_ri/   I think what happened was that these directories did have files in them, but then a developer realized he/she shouldn't have checked them in in the first place since these are by-products of the build process, so he/she removed the files but somehow the empty directories are left in the repository as ancient relics.  How can I remove this from CVS? The only results I seem to be able to find on google say that there shouldn't be a need to remove empty directories as CVS won't keep them around in the first place, and that the -P (prune) options to cvs update should remove them from the working directory  - which is zero help if you actually have empty directories in your repository.  A cvs remove and cvs commit doesn't seem to take care of this situation:  $ cvs remove -Rf com cvs remove: Removing com cvs remove: Removing com/companyname cvs remove: Removing com/companyname/blah cvs remove: Removing com/sun cvs remove: Removing com/sun/java cvs remove: Removing com/sun/java/jax_rpc_ri $ cvs commit com cvs commit: Examining com cvs commit: Examining com/companyname cvs commit: Examining com/companyname/blah cvs commit: Examining com/sun cvs commit: Examining com/sun/java cvs commit: Examining com/sun/java/jax_rpc_ri $ ls -l com total 24 drwxrwxr-x  2 matt matt 4096 Oct 15 14:38 CVS drwxrwxr-x  9 matt matt 4096 Oct 15 14:38 companyname drwxrwxr-x  4 matt matt 4096 Oct 15 14:38 sun   It's still there!  Does SVN have this weird behavior too? How can I delete Facebook messages I've sent to other Facebook users from their inbox?  I have had a conversation with my friend but now I want to delete my messages from that conversation. Can I have them deleted from my friend's inbox? Reason behind this self invoking anonymous function variant Enabling CORS in GeoServer (jetty)? Linux genisoimage Batch Android "M" was officially announced on at the Google I/O conference on May 28, 2015 and revealed to be 6.0 Marshmallow on August 17, 2015.  You can review the high-level changes at the  and the announcement on the official .  Of course, not all devices will get this update right away and others won't get it at all. Each manufacturer and carrier often choose to add their custom modifications which tend to delay the official release.  When will each device get Android 6.0, either officially or unofficially, if ever?  See also: finding rational roots of polynomials Could someone please explain me how to apply the Descartes's Criterion? For example , how do I find the rational roots of $ x^3 -x +1$.  I've been looking at some examples, but I get confused. Google Chrome is my favorite browser, and I can't imagine going on without it. Suddenly (without closing and re-opening) the browser; chrome has been reading characters upside down, very weird:    Chrome version:    I tried restarting my browser and my MAC (running OS X Yosemite 10.10), but the error keeps on coming. A field that is conservative must have a curl of zero everywhere. However, I was wondering whether the opposite holds for functions continuous everywhere: if the curl is zero, is the field conservative? Can someone please give me an intuitive explanation and insight into this, and if it is true, why? Also, please try to not be too rigorous (only in grade 9). I need to show that for any countable ordinal $\alpha$ there is a set A $\subseteq \mathbb{Q}$ such that (A, &lt;) is isomorphic to ($\alpha, \in$).  To do it I am supposed to show the following stronger statement by induction on ordinals $\alpha$ &lt; $\omega_{1}$:  Let P($\alpha$) be the statement "For every interval (a, b) with rational endpoints, there is an A $\subseteq$ (a,b) such that (A, &lt;) is isomorphic to ($\alpha, \in$)."  I think that the purpose of breaking $\mathbb{Q}$ into intervals (a,b) is to make $\mathbb{Q}$ a countable set, but I don't see the significance of doing this or how it would make the proof easier.  I feel like I need to define a well-ordered set in (a,b) but I don't know how to do this since if a and b are both negative then there is no least element, so no well-ordering.  I believe that I am supposed to be using transfinite induction? But I don't know how P($\alpha$) implies P(S($\alpha$)) for S($\alpha$) successor of $\alpha$.  Any help would be greatly appreciated... thanks in advance. Given: a = 1 b = 10 c = 100  How do I display a leading zero for all numbers with less than two digits? This is the output I'm expecting: 01 10 100 Activity restart on rotation Android I want to convert a std::string to lowercase. I am aware of the function tolower(). However, in the past I have had issues with this function and it is hardly ideal anyway as using it with a std::string would require iterating over each character. Is there an alternative which works 100% of the time? I am trying to show that if $n$ is an odd natural number, then $8$ divides $n^{2}-1.$ I was able to prove that because I know that if $n$ is an odd natural number, then $n^{2}$ can be written as $8k+1$ for some $k\in \mathbb{N}.$ I would like to show this question by using the Euclidian division. Then I wrote $n^{2}-1=8k+r$, where $0\leq r  &lt; 8.$ When $r=2$ we get $n^{2}-1=8k+2.$ Since $n$ is odd, then $n^{2}-1$ is even and I got stuck. Is there way to fix that? My new laptop came preinstalled with Windows 8. Naively, as I am, I just formatted the harddrive and installed fine old Ubuntu. Now I want to install Windows 8 for dual boot again, but I have no DVD and to download the ISO one needs a product key. That key is not on the back of the laptop anymore but somewhere on the mainboard.  Is there any way to recover the product key from the mainboard using Ubuntu? Proving that an additive function $f$ is continuous if it is continuous at a single point Suppose that $f$ is continuous at $x_0$ and $f$ satisfies $f(x)+f(y)=f(x+y)$. Then how can we prove that $f$ is continuous at $x$ for all $x$? I seems to have problem doing anything with it. Thanks in advance.