It is well known that the sum of all inverse primes is divergent. But the alternating sum is convergent by the Leiniz criterion. To which known constant "a" does the sum converge?  $$a = \frac{1}{2} - \frac{1}{3} +\frac{1}{5}-\frac{1}{7}+\frac{1}{11} -+ ...$$
$$\sum_{n = 1}^\infty \frac{(-1)^n}{p_n}$$ where $p_n$ is the $n$th prime.  I have computed this to 10000 rather than infinity. My results suggest that convergence does happen but it's very slow. But I can't even be sure about the first few digits: $-0.26959$?  I have looked at the "questions that may already have your answer" and some of the "similar questions," but they involve somewhat different formulas.
If G is not commutative, then is there always a subgroup that is not a normal subgroup? Can there be only one extension to the factorial? Usually, when someone says something like $\left(\frac12\right)!$, they are probably referring to the Gamma function, which extends the factorial to any value of $x$.  The usual definition of the factorial is $x!=1\times2\times3\times\dots x$, but for $x\notin\mathbb{N}$, the Gamma function results in $x!=\int_0^\infty t^xe^{-t}dt$.  However, back a while ago, someone mentioned that there may be more than one way to define the factorial for non-integer arguments, and so, I wished to disprove that statement with some assumptions about the factorial function.    the factorial   is a  for $x\in\mathbb{C}$ except at $\mathbb{Z}_{&lt;0}$ because of singularities, which we will see later. is a monotone increasing function that is concave up for $x&gt;1$. satisfies the relation $x!=x(x-1)!$ and lastly $1!=1$     From $3$ and $4$, one can define $x!$ for $x\in\mathbb{N}$, and we can see that for negative integer arguments, the factorial is undefined.  We can also see that $0!=1$.  Since we assumed $2$, we should be able to sketch the factorial for $x&gt;1$, using our points found from $3,4$ as guidelines.  At the same time, when sketching the graph, we remember $1$, so there can be no jumps or gaps from one value of $x$ to the next.  Then we reapply $3$, correcting values for $x\in\mathbb{R}$, since all values of $x$ must satisfy this relationship.  Again, because of $1$, we must re-correct our graph, since having $3$ makes the derivative of $x!$ for $x\in\mathbb N$ undefined.  So, because of $1$ and $3$, I realized that there can only be one way to define the factorial for $x\in\mathbb R$.  Is my reasoning correct?  And can there be only one extension to the factorial?    Oh, and here is a  to how I almost differentiated the factorial only with a few assumptions, like that it is even possible to differentiate.  Putting that in mind, it could be possible to define the factorial with Taylor's theorem? Was Voldemort allowed to visit Hogsmeade while in school? We know that in order for third-years and up to be allowed to visit the village of Hogsmeade, they must present a form duly signed by their parent or guardian. When Harry’s guardians, the Dursleys, miss their opportunity (!) to sign his, he is not allowed to go to Hogsmeade. Both Cornelius Fudge (as Minister for Magic) and Professor McGonagall (as head of Gryffindor House) refuse to sign the form on his guardians’ behalf—probably at least in part because they think there’s a mad killer on the loose looking for Harry, but also, as McGonagall specifically states, simply because:  “You heard what I said. No form, no visiting the village. That’s the rule. […] The form clearly states that the parent or guardian must give permission.” Prisoner of Azkaban, Bloomsbury/UK paperback, p. 113  Obviously, Tom Riddle did not have any parents or guardians ready to sign his form for him: his mother was dead, his father likely did not even wish to know of his existence, and if he did have a legal guardian at all (I don’t know how orphanages worked in Britain in the 1930s), it was likely to be  (warning: Wikia link) as the matron of Wool’s Orphanage where Riddle grew up. As Mrs. Cole was perfectly Muggle in every way and was never told by Dumbledore—nor, we must assume, by the boy himself—that Riddle was a wizard attending a wizardry school, it seems unlikely in the extreme that she ever signed Riddle’s form to visit Hogsmeade.1 I don’t recall any incidents in the books where Voldemort speaks specifically of him visiting Hogsmeade as a school boy (then again, why would he?); but being content to remain at the castle while his classmates have fun in the village seems quite out of character for him, too. Are there any (canonical) indications whether special provisions were made for orphans with no legal guardians to sign, so that they were allowed to visit Hogsmeade after all? Or, oppositely, whether such cases were played strictly by the book and unguardianed orphans, and thereby Riddle, were not allowed to visit Hogsmeade?    1 One might of course easily envision that Riddle simply Confounded Mrs. Cole to sign the form; but that would lead to detection of underage magic (assuming this was even a thing in 1940). Riddle may of course have forged Mrs. Cole’s signature, but that would be rather likely to be discovered as a forgery, since at least Dumbledore was aware that Mrs. Cole did not know of Riddle’s magical abilities. Can I apply for two Schengen visas if I have two subsequent trips within a short period? I am traveling to Germany on May 1st for an exhibition for which I have already obtained a visa and its validity is up to May 13th. I am returning on May 12th and then traveling on May 17th to UK, and from UK to Switzerland for vacation with family. As time is very short for applying for a Schengen visa again after returning on the 12th with May 13-14 being a Saturday and Sunday, can I apply for Schengen visa again and can I get 2 valid Schengen visa? An infinite series plus a continued fraction by Ramanujan Here is a famous problem posed by Ramanujan     Show that $$\left(1 + \frac{1}{1\cdot 3} + \frac{1}{1\cdot 3\cdot 5} + \cdots\right) + \left(\cfrac{1}{1+}\cfrac{1}{1+}\cfrac{2}{1+}\cfrac{3}{1+}\cfrac{4}{1+\cdots}\right) = \sqrt{\frac{\pi e}{2}}$$   The first series seems vaguely familiar if we consider the function $$f(x) = x + \frac{x^{3}}{1\cdot 3} + \frac{x^{5}}{1\cdot 3\cdot 5} + \cdots$$ and note that $$f'(x) = 1 + xf(x)$$ so that $y = f(x)$ satisfies the differential equation $$\frac{dy}{dx} - xy = 1, y(0) = 0$$ The integrating factor here comes to be $e^{-x^{2}/2}$ so that $$ye^{-x^{2}/2} = \int_{0}^{x}e^{-t^{2}/2}\,dt$$ and hence $$f(x) = e^{x^{2}/2}\int_{0}^{x}e^{-t^{2}/2}\,dt$$ Thus the sum of the first series is $$f(1) = \sqrt{e}\int_{0}^{1}e^{-t^{2}/2}\,dt$$ But I have no idea about the continued fraction and still more I am not able to figure out how it would simplify to $\sqrt{\pi e/2}$ at the end.  Please provide any hints or suggestions.  Update: We have $$\begin{aligned}f(1) &amp;= \sqrt{e}\int_{0}^{1}e^{-t^{2}/2}\,dt = \sqrt{e}\int_{0}^{\infty}e^{-t^{2}/2}\,dt - \sqrt{e}\int_{1}^{\infty}e^{-t^{2}/2}\,dt\\ &amp;= \sqrt{\frac{\pi e}{2}} - \sqrt{e}\int_{1}^{\infty}e^{-t^{2}/2}\,dt\end{aligned}$$ and hence we finally need to establish $$\sqrt{e}\int_{1}^{\infty}e^{-t^{2}/2}\,dt = \cfrac{1}{1+}\cfrac{1}{1+}\cfrac{2}{1+}\cfrac{3}{1+}\cfrac{4}{1+\cdots}$$ On further searching in Ramanujan's Collected Papers I found the following formula $$\int_{0}^{a}e^{-x^{2}}\,dx = \frac{\sqrt{\pi}}{2} - \cfrac{e^{-a^{2}}}{2a+}\cfrac{1}{a+}\cfrac{2}{2a+}\cfrac{3}{a+}\cfrac{4}{2a+\cdots}$$ and it seems helpful here. But unfortunately proving this formula seems to be another big challenge for me. What does 'Ign' , `Get` or 'Hit' mean when running an apt-get update? Where is .bash_profile? I want to put new aliases to my .bash_profile file, but I cannot find this file. Where is it supposed to be? Why does network ad URL from SO to history.SE mention Ask Ubuntu? Today, I was on Stack Overflow when I noticed an interesting ad pointing to history.SE (actual square ad, not just Hot Network Questions).  The link in the ad was:    which resolves to:  https://history.stackexchange.com/questions/23946/what-were-greek-and-roman-sails-made-from?utm_medium=ad&amp;utm_source=http-askubuntu-com-questions-471178-converting-hundreds-of-jpg-to-pdf-using-te&amp;utm_campaign=history.stackexchange.com-promo&amp;utm_content=23946  (link form for conveniece: )  The confusing part os the URL is: utm_source=http-askubuntu-com-questions-471178-converting-hundreds-of-jpg-to-pdf-using-te. The ad is from SO to history.SE, it doesn't have anything to do with Ask Ubuntu, so why does it appear in the URL? Is this a bug?  EDIT: Okay, managed to trigger it again: the page  contains this ad:    The link of the ad is:    which resolves to:  http://opensource.stackexchange.com/questions/1556/how-to-react-to-unhelpful-contributions-to-otherwise-unnoticed-projects?utm_medium=ad&amp;utm_source=http-sharepoint-stackexchange-com-questions-16925-why-am-i-getting-server-erro&amp;utm_campaign=opensource.stackexchange.com-promo&amp;utm_content=1556  incorrectly referencing an unrelated sharepoint.SE question. What is the difference between "retired out" and "retired not out" in cricket? How to prove that $P \rightarrow Q$ is equivalent with $\neg P \lor Q $? In my book about Logic, which is called 'Language, Proof and Logic', by the way, there is explained that the conditional $ P \rightarrow Q $ is equivalent with $\neg P \lor Q$.   that gives as answer: 'just compare the truth tables and you can see that they are equivalent'. However, I would like to know how to prove this using formal propositional logic. This should be possible, right? Where do I find out how a command/environment is defined? Say I want to redefine the itemize environment, or make a modified version of it. I want to know how it is currently defined. Where do I find this information? If it's some particular package command, I can just look at the .sty file for that package, but if it's something more basic, I don't know where to look...  Is there a reference manual of where all this stuff is explained? Or is there a file in the depths of my texmf tree I should poke around in? A lot of strange loop devices appear in my Kubuntu OS.      [translation:     Hard disk 107.4GB     Hard disk 487.0GB     Loop Device     Hard Disk 4.0GB     Loop Device     Loop Device     Loop Device     Loop Device     Victor Disk]   Can anybody tell me what it is? . How do you compute negative numbers to fractional powers? How can I get the DateTime for the start of the week? If I'm given a complex number (say $9 + 4i$), how do I calculate its square root? What are beginner’s books on typography? What are the best books to study Neural Networks from a purely mathematical perspective? I am looking for a book that goes through the mathematical aspects of neural networks, from simple forward passage of multilayer perceptron in matrix form or differentiation of activation functions, to back propagation in CNN or RNN (to mention some of the topics).  Do you know any book that goes in depth into this theory? I've had a look at a couple (such as Pattern Recognition and Machine Learning by Bishop or Deep Learning by Goodfellow, Bengio and Courville) but still have not found a rigorous one (with exercises would be a plus). Do you have any suggestions? Born's Rule, What is the Reason? How do I use filenames that start with a dash "-" as command arguments? Noticed in :         is 320, shown 322. Also locked in 2013.  And then :         has a score of 179, shown 180.  has a score of 123, shown 122.  Neither of these posts got any votes recently.