How to determine the sum of an infinite series that is neither arithmetic nor geometric?
Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$?
What is the proper use of [square brackets] in quotes? I've read a great article about the . But there are still some points that are unclear.  Why do we say...   school project but not school's project? car service but not car's service or even cars' service (plural form)? apostrophe usage but not apostrophe's usage or apostrophes' usage? How to apply a macro to each column of a table Collinearity diagnostics problematic only when the interaction term is included Interpreting the residuals vs. fitted values plot for verifying the assumptions of a linear model Consider the following figure from Faraway's Linear Models with R (2005, p. 59).    The first plot seems to indicate that the residuals and the fitted values are uncorrelated, as they should be in a homoscedastic linear model with normally distributed errors. Therefore, the second and third plots, which seem to indicate dependency between the residuals and the fitted values, suggest a different model.  But why does the second plot suggest, as Faraway notes, a heteroscedastic linear model, while the third plot suggest a non-linear model?  The second plot seems to indicate that the absolute value of the residuals is strongly positively correlated with the fitted values, whereas no such trend is evident in the third plot. So if it were the case that, theoretically speaking, in a heteroscedastic linear model with normally distributed errors  $$ \mbox{Cor}\left(\mathbf{e},\hat{\mathbf{y}}\right) = \left[\begin{array}{ccc}1 &amp; \cdots &amp; 1 \\ \vdots &amp; \ddots &amp; \vdots \\ 1 &amp; \cdots &amp; 1\end{array}\right] $$  (where the expression on the left is the variance-covariance matrix between the residuals and the fitted values) this would explain why the second and third plots agree with Faraway's interpretations.  But is this the case? If not, how else can Faraway's interpretations of the second and third plots be justified? Also, why does the third plot necessarily indicate non-linearity? Isn't it possible that it is linear, but that the errors are either not normally distributed, or else that they are normally distributed, but do not center around zero? How does sudo handle $HOME differently since 19.10? Easiest way to prove that $2^{\aleph_0} = c$ $\aleph_0$ is the cardinality of the set of natural numbers, $\aleph_0 = |N|$. $c$ is the cardinality of the continuum, i.e. the set of real numbers $c = |R|$.  I know that $|P(A)| = 2^{|A|}$. This means that the cardinality of the power set of a set is 2 raised to the power of the cardinality of that set.  This basically means that to prove  $2^{\aleph_0} = c$, I need to prove $c = |P(N)|$. I've seen the Cantor diagonal argument, although I have no idea on how to use it to prove this.   Could anyone suggest ideas of the simplest way to prove that $2^{\aleph_0} = c$? Labelled spanning trees of $K_n-e$ Let $e$ be an edge of $K_n$- the complete graph on $n$ vertices. Prove that the number of labelled spanning trees of $K_n-e$ is $(n-2)n^{n-3}$.  I think the answer lies in using some modified form of Prüfer's sequence but I can't quite nail it. Windows & Ubuntu dual boot.No menu to choose OS? From time to time certain chat rooms (JavaScript and probably PHP, too) attract the attention of a fellow help vampire who thinks he can start sucking from the people in there.  Often their message end up being flagged as offensive (and those flags are likely to be validated, too, because people are annoyed by such behaviour) but it's kind of an inappropriate use of those flags.  I'm proposing a new chat flag type, which is especially meant for help vampires. Flagging it would present the flag to 10k users and owners of the room where the flag originated (no matter how much rep they have). When the flag is validated it would cause messages from this user to be shown with a low opacity for maybe three hours. It could also notify the user that help vampirism is a bad thing and that he needs to improve his behaviour.  Repeated flags of this type - obviously with some time in between in case multiple message which are really part of the same conversation are flagged - should then result in a temporary ban from the room where the messages got flagged. If that doesn't help the next step would probably be a temporary ban from all chat rooms.    For clarification, I'm not talking about the quick "this line doesn't work, why?" stuff but the kind of help vampires who get cocky even though they just got e.g. a link to a SO question which has a perfectly fine and fitting answer. Because hey, you didn't give them teh codez! I'm using JUnit 4.4 and Maven and I have a large number of long-running integration tests.  When it comes to parallelizing test suites there are a few solutions that allow me to run each test method in a single test-class in parallel. But all of these require that I change the tests in one way or another.  I really think it would be a much cleaner solution to run X different test classes in X threads in parallel. I have hundreds of tests so I don't really care about threading individual test-classes.  Is there any way to do this? Given 3 distinct primes {$p,q,r$}, then $|G|=pqr \implies G$ not simple How can I get a copy of a Canadian visa refusal letter? I have applied for a Canadian Temporary Resident Visa (TRV); a previous paper application was refused in 2016. I have lost the copy of my refusal letter and want to submit it when I apply for my new visa. How how can I get the a copy of the 2016 refusal letter from Embassy? Counting lines, words, characters and top ten words? Terminal appears with much delay when shortcut is used When I press Ctrl+Alt+T the Terminal opens after around 30 seconds. That's pretty annoying. If I open it with Alt+F2 and enter gnome-terminal the window appears instantly as usual.  I'm using Ubuntu 16.04 with gnome3.  Thank you for your help in advance! How to pick a specific Pantone color in Photoshop? What about finding a Pantone that is close to a specific CMYK color? How would I select a specific Pantone color in Photoshop, e.g. PMS 5625?  Also, how do you pick a Pantone colour that is similar to a specific CMYK color, e.g. C61, M41, Y58, K16 green?    In this case Pantone 5615C looks closest to the CMYK color. $$ A $$  Light is travelling away from the absolute center of the Schwarzchild black hole (i.e. none of that spinning/stretching field) at $\ C $, and so shouldn't $\ \Delta C$ need to be greater than $\ C $ in order for the light to collapse into the black hole, therefore violating the relativistic limit of $\ C $?    It seems like light would just 'hover' right in the center of a black hole then. Because if it goes away from it, the event horizon pulls it back, and if it goes into it, the velocity change must be greater than $\ C $ which violates general relativity; so technically there's no way for the light to move away, and so it would just be still.  The thing is that the light must move at the speed of light or general relativity doesn't work. It's massless... But in the center of a black hole, if the above is true (it can't move or escape), then it is not travelling at the speed of light. So it cannot be still.  It cannot be spinning in a constant orbit either, because that would require velocity in an outward angle. And it cannot move in an outwardly fashion, only in an inwardly one.  Is it spinning in an infinitely decaying spiral towards the center?  If so what happens to it if and when it reaches the direct and complete center, i.e. The case of it being 'shot' from the center.    $$ B $$  If you shot light at an angle towards a black hole, but not directly to the center of it  then that light from the outside will eventually become the same as the light that I said was shot from the center to the outside; because in both cases, they end up in an infinitely decaying spiral.    If so it would end up exactly like the light in the above paragraph, so then we need to figure out what happens to that light to figure this one out, or vice versa.    $$ C $$  If you shot light directly towards the center of a black hole it would have no spiral to decay into, so therefore it must stop at some point in the center of a black hole. If it stops though, it loses it's momentum and violates general relativity again because it's energy "disappears".    If so then it violates conservation laws. So I'm assuming this simply is not true.    Is the light travelling into a black hole somehow still travelling at the speed of light, in every direction possible where that direction keeps it facing towards the center?  If so, why is there not a point at which the only thing left for the light to do is stay still, since all other paths would increase the distance between it and the black hole's center, and therefore become Case A. And if there isn't, what would happen if there was?  Also, if it's a time and observation answer where it supposedly takes and infinite amount of time for C to hit the center, wouldn't it be the case that the black hole is 'eating' all of the available light in the universe and keeping it in stasis?  Also, what happens in a Kerr black hole then? Does it just rip the field of time/gravity at an angle such that the light would never get closer than the center? Reducing Lion OS Footprint by removing unnecessary files/folders I am trying to downsize my iMac 640Gb disk to make it fit onto a 120Gb (113Gb usable) SSD.  The total size of use files is currently approximately 340Gb, of which my home folder accounts for 265Gb (which I intend to not put on the SSD, or at least not all of it).  Very rough calculations suggest that my OS and application files are taking up roughly 75Gb, which puts me approximately 2/3rds full - not bad.  But, I would like to reduce this a little further if possible, to ensure that as and when I continue to use the computer (and I have a number of large apps that do not currently have installed including music production apps with a lot of data) I am not worried about running out of space on the boot drive.  To this end I already own XSlimmer which I will use periodically to remove PowerPC and language variants from my apps, but I was wondering if there are any areas of the Base OS that I can safely remove to save further space?  I am thinking about things like OS localisation, the location of the included screensavers and wallpapers, the voice files that are used for text-to-speech etc.  Is there a significant amount of such data that can be easily removed to save a few extra Gb on the boot disk? How to cross-reference an inequality with \cref? I have inequalities being cross-referenced by \cref as equations.  1-) How can I change that? (and still have inequalities with the same numbering as equations)  2-) How can I change that and have inequalities with their own numbering?  \documentclass[12pt]{article} \usepackage{amsmath} \usepackage{cleveref} \usepackage[letterpaper]{geometry} \begin{document}  \begin{flalign} &amp;f(x) \geq g(x)&amp; \label{i:positivity} \end{flalign}  Admitting that \cref{i:positivity} has a solution, ...    \end{document} I have a COM dll written in vb6. When I try to create a new object of a class module from this dll I get a Run timer error 430: Class does not support automation or does not support expected interface. The interesting thing is that this happens only from outside the IDE, when I am debugging from within the IDE there is no error thrown and the new object of the class is created successfully. What can be the cause?  In general I occasionally get these kind of errors in COM dlls. What is the best way to debug COM issues? How can I know the path of the dll that is being used when a program is running?