Let $N\geqslant1$ be an integer. Then which of the following statements are true ?  $\displaystyle\sum\limits_{n=1}^N\dfrac1n\leqslant1+\log N$  $\displaystyle\sum\limits_{n=1}^N\dfrac1n&lt;1+\log N$  $\displaystyle\sum\limits_{n=1}^N\dfrac1n\leqslant\log N$  $\displaystyle\sum\limits_{n=1}^N\dfrac1n\geqslant\log N$   My Attempt : $$\underbrace{\frac11+\frac11+\frac11+\frac11+\frac11+\frac11+\ldots+\frac11}_{N\text{ times}}=N$$ $$\geqslant\sum_{n=1}^N\frac1n=\frac11+\frac12+\frac13+\frac14+\frac15+\frac16+\ldots+\frac1N$$ $$\geqslant\underbrace{\frac1N+\frac1N+\frac1N+\frac1N+\frac1N+\frac1N+\ldots+\frac1N}_{N\text{ times}}$$ $$=\frac NN=1$$ $$\implies1\leqslant\sum_{n=1}^N\frac1n\leqslant N$$ Also $\;\log N\leqslant N,\;\forall N\geqslant1\,.\;$ I'm unable to combine this information to answer the question. Please help me.
Consider the partial sum of the divergent Harmonic series $H_n = \sum\limits_{k = 1}^{n}\frac{1}{k}$. I recently saw a question which required finding out the asymptotic bounds of $H_n$. Now, I could not find any bound closer than $O(n)$. So I looked up the series and found  Wikipedia article which says that $H_n - \ln n$ approaches a constant (the ). This clearly shows that $H_n = \Theta(\log n)$.  However, considering the question was in an beginners undergraduate algorithms course in CS, I was wondering if there is a simpler proof of this? Is there some clever proof requiring elementary mathematics to show $H_n = \Theta(\log n)$?  PS: This is not homework in the sense that it is not my homework, and the actual question does not ask for a proof, but rather asks a lot of functions to be arranged asymptotically.
window.onbeforeunload and window.onunload is not working in Firefox, Safari, Opera? A couple years ago I ran upon a  video demonstrating how researchers used x-rays given off by tearing tape off its spindle in hopes to miniaturized and cheapen future x-ray devices. As of late I have been wondering what interactions are causing this. I understood from the video that the electrons are being ejected due to the added energy to the system (peeling). This instinctively pointed me towards QED where interactions between electrons produce a photon for short periods. 1)Are those same photons from electron interaction the resulting x-rays? 2)Are the frequencies of the photons being limited by the $\Delta$energy$\Delta$time uncertainty, and/or $\Delta$position$\Delta$momentum uncertainty (at the vertex of the ripping tape, where high densities of electrons may accumulate) ? Note: I've also considered that the van der Waals force may come into consideration, not certain though. Looking for a QED based descriptions/answers. Any elementary references to this work is appreciated. Possible Duplicate:             What is correct? The jury was divided or The jury were divided? I am told that the latter is accurate because all of the jury do not have the same opinion in the particular case. Then, how does the subject verb agreement fit in 'The jury is still out' ? I mean, 'The jury is out' would allude to a situation were the opinion on a matter is (are?) still divided. Note: This is not the same as , which is about the moderator tool. I am talking about the flag history page that is accessible through the user profile at .  I tend to have a pretty good hit rate for flags, well above 90% helpful on several sites. I also semi-regularly review the flag history page, both to see the status of any outstanding flags as well as see how flags I raised have been handled. Particularly, whenever a flag of mine is declined, I look over the post I raised the flag against with this new information in mind, to try to determine if I should change my flagging behavior in some way to better match the specific site's scope.  However, comment flags are difficult in this regard. Most often, there is only a binary status: helpful or declined. Helpful comment flags nearly always lead to the comment being deleted, so that obviously could present a bit of a problem, but that's not really relevant because a helpful flag means you should just keep doing what you've been doing. Declined comment flags, on the other hand, usually mean the comment is kept. It's the declined comment flags that I want to learn more about. But there's no way to!  All the flag says is which post it was on, how I flagged a comment, that I was the one to raise the flag (which frankly is rather useless information when you are already looking at your own flags summary), when the flag was raised and whether it's pending, helpful or declined. In a very few cases the moderator handling the flag may leave a helpful return comment.  This can get even more confusing if I raise multiple flags in quick succession against comments on the same question including any answers to it, because they basically all get bundled together.  Even in the case of a declined flag where the comment remains, there is no way to find your way back to the specific comment you flagged! Thus it is very difficult to recognize the error of your ways. Because comment flags (with few exceptions) are handled by diamond moderators only, the lack of feedback about which comment's flag was declined means it's easy to keep wasting the moderators' time without even intending to do so.  Hence my feature request: add a link to the specific comment that the specific flag was raised against, ideally only if that comment remains. Ubuntu 20.04 on DualBoot fail 90% of tries - Initramfs unpacking failed: Decoding failed I just installed Ubuntun 20.04 on a Dell Latitude 7400 on DualBoot. I had to remove the RST as mentioned on this post :  It works but now I need to test 5 or more times to boot. After the grub screen, most of the time i have a black screen. Sometimes I have this error : Initramfs unpacking failed: Decoding failed And then it starts.  here is a file called initramfs.conf cat /etc/initramfs-tools/initramfs.conf MODULES=most  BUSYBOX=auto  COMPCACHE_SIZE=&quot;&quot;  COMPRESS=lz4   DEVICE=  NFSROOT=auto RUNSIZE=10% How to search the internet for terms with special characters As I learnt from this , Google has no ability to search for special characters. Let's say I want to search for ss!. Are there any other search engines that can achieve this? Please do not propose any programming code search engines since this is a general search. How would I go about identifying a port, card or connector on my system which I am unfamiliar with?  Where would I find information on what these parts are? What would be the steps I would take in identifying a unknown port or cable?   This question comes up frequently, and the suggested solutions are usually the same. This community wiki is an attempt to serve as the definitive, most comprehensive answer possible. Feel free to add your contributions via edits. In my special package combination I get the error No room for new \write. \tablecontents and Bad number (16). \tablecontents. The following MWE throws this error in pdflatex and lualatex with TeX Live 2015. As you can see, I add the own content register loi. I can delete nearly every package separately and the error directly vanishes.  Does anyone have an idea what is wrong with this package combination?  \documentclass{scrartcl} % load packages \usepackage{ifluatex} \ifluatex     \usepackage{fontspec} \fi \usepackage{adjustbox, amsmath, blindtext, datatool, fancybox, fixme, graphicx, marginnote, pgfplots} \usepackage{imakeidx} \usepackage{biblatex} \usepackage{glossaries} \usepackage{hyperref} % load package for code code highlighting \usepackage{minted}  % define Open Issue and corresponding List of Open Issues \newcommand{\openissue}[2][Unlabeled]{%     \par%     {\refstepcounter{loicounter}%     \phantomsection% comment out if hyperref is noy used     \addcontentsline{loi}{figure}{%         \protect\numberline{%             \ifcsname c@chapter\endcsname%                 \thechapter.%             \fi%             \theloicounter%         }{#1: #2}%     }%     \textbf{Open Issue -- #1:} #2}%     \par% } % define counter \makeatletter \ifcsname c@chapter\endcsname%     \newcounter{loicounter}[chapter]% \else     \newcounter{loicounter}% \fi% \newcommand*{\listopenissuesname}{List of Open Issues} \newcommand{\listofopenissues}{%     \ifcsname chapter\endcsname%         \chapter*{\listopenissuesname}%     \else%         \section*{\listopenissuesname}%     \fi%     \@starttoc{loi}% } \makeatother  \fxsetup{status = draft} \makeglossaries \makeindex  \begin{document}  % include open issues \listofopenissues % include list of corrections \listoffixmes % include indicies \tableofcontents                            % Inhaltsverzeichnis einbinden  \end{document} OS X Yosemite brought support for recording iPhone/iPad screen natively without having to using AirPlay applications like AirServer or Reflector.   How do I record the iPhone screen on macOS ? Definite integral, quotient of logarithm and polynomial: $I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$ I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{\lambda ^2x^2+\lambda x+1}\text{d}x =\frac{1}{\lambda}\int_0^{\infty}\frac{(\ln x-\ln \lambda)^2}{x^2+x+1}\text{d}x \\ =\frac{\ln ^2\lambda}{\lambda}\int_0^{\infty}\frac{1}{x^2+x+1}\text{d}x+\frac{1}{\lambda}\int_0^{\infty}\frac{\ln ^2x}{x^2+x+1}\text{d}x \\ =\frac{\ln ^2\lambda}{\lambda}\frac{2\pi}{3\sqrt{3}}+\frac{2}{\lambda}\int_0^1\frac{\ln ^2x}{x^2+x+1}\text{d}x$$ But I hav no idea on process the remaining integral:( Anyone knows how to solve it?  THX guys! I came with another method might work for this: Recall the handy GF $$\frac{1}{x^2+x+1}=\frac{2}{\sqrt{3}}\sum_{k=0}^{\infty}\sin \left(\frac{2\pi}{3}\left(k+1\right)\right)x^k$$ Then we have :  $$\int_0^1\frac{\ln ^2x}{x^2+x+1}\text{d}x=\frac{2}{\sqrt{3}}\sum_{k=0}^{\infty}\frac{2}{\left(k+1\right)^3}\sin \left(\frac{2\pi}{3}\left(k+1\right)\right)x^k$$ With $$\sum_{k=1}^{\infty}\frac{\sin (kx)}{k^3}=\frac{\pi ^2}{6}x-\frac{\pi}{4}x^2+\frac{x^3}{12}$$ What can we arrived? The beamer command \alt&lt;*overlay specification*&gt;{foo}{bar} will insert foo if the current slide is within the given overlay specification, and bar otherwise.  It's equivalent to   \only&lt;*overlay specification*&gt;{foo}\only&lt;*complementary overlay specification*&gt;{bar}   Here is a sample document using it:  \documentclass{beamer} \begin{document} \begin{frame} \begin{align*} y &amp;= \frac{(x^2+1)\sqrt{x+3}}{x-1} \\ \ln y &amp;= \ln (x^2+1) + \frac{1}{2} \ln(x+3) - \ln(x-1) \\ \frac{1}{y} \frac{dy}{dx}       &amp;= \frac{2x}{x^2+1} + \frac{1}{2(x+3)} - \frac{1}{x-1} \end{align*} So \[ \begin{split} \frac{dy}{dx} &amp;= \left(\frac{2x}{x^2+1} + \frac{1}{2(x+3)} - \frac{1}{x-1} \right) \alt&lt;2&gt;{\frac{(x^2+1)\sqrt{x+3}}{x-1}}{y} \end{split} \] \end{frame} \end{document}   This is very useful and saves typing.  But if one of the alternatives is larger than the other, it leads to jiggling slides.    What I would like is a overlay-specification-aware command \altvisible that would take the same amount of space on each slide.  So it would have to insert a box as big as the bigger of the two, and then set the correct material.  So the above document with \altvisible instead of \alt would not jiggle.  Any takers? Is there an official Marvel Cinematic Universe viewing order? t-test for partially paired and partially unpaired data NewForm.aspx disappeared The "" formulas often come in handy, but it's not immediately obvious that they would be true.  \begin{align} \sin(\alpha \pm \beta) &amp;= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ \cos(\alpha \pm \beta) &amp;= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \end{align}  So what I want to know is,    How can I prove that these formulas are correct? More importantly, how can I understand these formulas intuitively?   Ideally, I'm looking for answers that make no reference to Calculus, or to , although such answers are still encouraged, for completeness. How to reference the caller object ("this") using attachEvent $(\frac 3p)$ is the Legendre symbol here, not a fraction, if that wasn't clear.  This is what I have so far:  We know by Gauss's Lemma that $(\frac qp) = (-1)^v$ where $$v = \#\left\{1 \le a \le \frac{p-1}{2} \mid aq \equiv k_a \pmod{12}),\space -p/2 &lt; k_a &lt; 0\right\}$$  And we also know that multiples of $(\dfrac{p-1}{2}) \times 3$ must be between $0$ and $p/2$, between $p/2$ and $p$, or between $p$ and $3p/2$.   Representing $p$ as $12k + r$ seems like the next step, but I'm not sure where to go from there. I have GM'd several RPGs before that are not D&amp;D (Fate, Deathwatch, Star Wars). We first tried to play D&amp;D using the 3.5 edition and it went horribly — too many books and too many options with no idea of where to start.  I have been told that 5E takes care of all that and is easier and more fun so I want to try that but I would like a definitive list of what I need.  I expect that the kinds of books we need divide up like this:   Rulebook for me as GM Rulebook for players Fluff for everyone Monster book for me as GM   Links to these would be helpful as we got very confused with the 3.5 books — who should use what books confused us.  What other things do we need to play a campaign? I expect things like:   Dice: what kinds and how many of each kind, roughly. Models and maps. Do we need these? Does 5E need models and a grid or can it be played without (which is preferred)?   Is there anything else that we need that I haven't anticipated? How do you handle aspect ratio differences with Unity 2D? I've gotten a lot of answers to this question, but they are all generic and generally not very useful.  None of the tutorials talk about aspect ratio and dealing with mobile devices and there are a zillion ways to do it, all seem to have gotcha's and flaws.  I really would love to know what successful games have used to handle different aspect ratios on iOS and Android without making a zillion different sized assets.  I am strictly speaking mobile, not desktop, specifically with Unity and I don't care about the UI, I only care about the gameplay canvas.  Issues I have in mind is when there are key things that have to be in certain places and cannot fall off the screen.  Using black bars on top or bottom is unacceptable these days. How does "warp drive" not violate Special Relativity causality constraints? I'm talking about this nonsense:   Now, I'm aware that there are problems with the practicalities (or possibilities) of methods involving exotic matter with negative mass, and that kind of thing has been addressed here:   But in this case my question is different and more specific. Sweeping all the other problems of constructing such a beast under the rug, wouldn't this violate causality?  SR says that the simultaneity and relative ordering of events is dependent on the relative motion of the observer, but that no matter how it works out, if event A causes event B, no observer will see B first. One of the many problems with FTL travel (or signals) is that if any information is sent faster than light, there will exist an inertial frame in which the signal arrives before it is sent, a blatant violation of causality. (I think we can all agree, can we not, that the threshold should be extremely high before breaking assumptions that we're in a causal universe?)  So how is any scheme to travel faster than light, even if it may locally satisfy the equations of GR, not a blatant violation causality according to SR?  Is my understanding of GR incomplete and there's something in it that obviates the causality implications of SR? Or am I correct that FTL claims should be dismissed with extreme prejudice on this basis alone? (Pending a serious replicable experiment showing causality to be breakable.)