I am out of hints here. Its trivial to show $1$ is a solution. How to show it's the only solution?  Can someone please give me some hints?  Please do not use congruence, limits or derivatives because they are out of scope of the chapter which has this question.
If $n&gt;1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer.  If you know , then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ appears in the sum, but $\frac{1}{2p}$ does not. Aside from $\frac 1p$, every other term $\frac 1k$ has $k$ divisible only by primes smaller than $p$. We can combine all those terms to get $\sum_{k=1}^n\frac 1k = \frac 1p + \frac ab$, where $b$ is not divisible by $p$. If this were an integer, then (multiplying by $b$) $\frac bp +a$ would also be an integer, which it isn't since $b$ isn't divisible by $p$.  Does anybody know an elementary proof of this which doesn't rely on Bertrand's Postulate? For a while, I was convinced I'd seen one, but now I'm starting to suspect whatever argument I saw was wrong.
Let me edit a post in the close review, then vote to close it I think we need a button that says Edit and Close added to the Review Close button options.  Many of these closed posts will be around forever (and should be, closed should not just be a path to deletion), so they need editing.  If I edit it now it goes away, with the only option being a back click to get back to the review, and then click into the post and edit it.  We have an Edit and Reopen button, so this would fit in just fine. Short-lived sci-fi series with a former astronaut turned superhero (or maybe a police officer?) using his spacesuit for his "powers" Plot Details/Summary I only watched an episode or two of this one, so I can't recall a whole lot about it.  The protagonist is a former astronaut of some sort, who makes it his mission to clean up the crime in his city.  He might have been a police officer in his &quot;new life&quot; or he might have decided to become a sort of quasi-superhero.  In either case, I distinctly recall his space suit being the gimmick that makes him an effective crime fighter in the show.  I seem to recall the protagonist being a young, dark-haired man, and I think he had a friendship or an alliance with a young woman (maybe a fellow cop, or someone from the space program?).  I don't think there were alien beings in the show.   The criminal threats were all quite terrestrial, IIRC. Timeframe and other Publication Details This was a syndicated tv show I saw on American television, probably around the late 90s to early 2000's.  I believe the show was filmed in Canada, though I'm not certain.  I don't think it lasted very long - it was pretty dreadful.  I can say for certain the show is not Space Cop or Gerry Anderson's Space Precinct. What is the most portable method of launching a terminal emulator? SQL server Read Only permission automatically for new databases My significant other and I finish our master degrees at the same time and thus we are applying for PhD programmes at the same time. One major factor of accepting a potential offer would be if we could study at the same university, so I feel that mentioning a two-body problem would be advantageous for both, us and the admission committee. But where do I mention this? Is a line in the statement of purpose suitable here?     If you admit me, please also consider admitting my significant other Jane Roe.   Some additional information: We both do research in computer science. While I do research in natural language processing my partner does research in computer vision. Generalized RBF Kernels There is the notion of Generalized RBF Kernels, for example in "Towards Optimal Bag-of-Features for Object Categorization and Semantic Video Retrieval" from Jiang (1) or in formula (2.72) in  (2).  They define: Gen. RBF Kernel: $e^{-pd(x,y)}$, where (1) says d can be any distance function and (2) requires it to be a metric.  I now have been wondering for a long time: Is a function defined by $e^{-pd(x,y)}$ where d is a metric always a Mercer's Kernel? I think in (2) it sounds like it, but if it were true, not many people know of it, since they are still looking to prove that $e^{-EMD(x,y)}$ is a kernel, where EMD = Earth Mover's Distance, which is a metric.  If its not a Mercer's Kernel - do you know of any counter example? Is it possible to add new alphabets to unicode-math? In ordinary LaTeX, one can easily add new math alphabets by using \DeclareMathAlphabet. One is then able to use, say, cmsy, eus, and rsfs letters side-by-side. Unfortunately, there does not seem to be a similar mechanism in unicode-math, or at the very least \DeclareMathAlphabet doesn't do what it's supposed to do.  Question. How can I set up my own math alphabets in unicode-math?   Attempted workarounds  Since unicode-math has lots of math alphabets, one workaround might be to replace an unused alphabet with the desired one using \setmathfont[range=...-&gt;...]. This doesn't quite work right in my example:  \documentclass[12pt]{article} \usepackage{fontspec,xltxtra,xunicode} \defaultfontfeatures{Mapping=tex-text}  \usepackage{unicode-math}  \scrollmode  \setmathfont{xits-math.otf} \setmathfont[range={\mathscr,\mathbfscr},StylisticSet=0]{xits-math.otf} \setmathfont[range={\mathcal,\mathbfcal},StylisticSet=1]{xits-math.otf} \setmathfont[range={\mathtt-&gt;\mathcal}]{latinmodern-math.otf}  \begin{document} \[ \begin{array}{c} \mathcal{A B C D E F G H I J K L M N O P Q R S T U V W X Y Z} \\ \mathscr{A B C D E F G H I J K L M N O P Q R S T U V W X Y Z} \\ \mathtt{A B C D E F G H I J K L M N O P Q R S T U V W X Y Z} \end{array} \]   I suspect the problem is that the codepoints of the calligraphic letters in Unicode are not contiguous.   Using \mathversion seems to work better:  \documentclass[12pt]{article} \usepackage{fontspec,xltxtra,xunicode} \defaultfontfeatures{Mapping=tex-text}  \usepackage{unicode-math}  \setmathfont{xits-math.otf}  \setmathfont[version=eucal]{latinmodern-math.otf} \newcommand{\eucal}[1]{\mathchoice %   {\mbox{\mathversion{eucal}$\displaystyle\mathscr{#1}$}} %   {\mbox{\mathversion{eucal}$\mathscr{#1}$}} %   {\mbox{\mathversion{eucal}$\scriptstyle\mathscr{#1}$}} %   {\mbox{\mathversion{eucal}$\scriptscriptstyle\mathscr{#1}$}}}  \setmathfont[range={\mathcal,\mathbfcal},StylisticSet=1]{xits-math.otf}  \begin{document} \[ \begin{array}{c} \mathcal{A B C D E F G H I J K L M N O P Q R S T U V W X Y Z} \\ \mathscr{A B C D E F G H I J K L M N O P Q R S T U V W X Y Z} \\ \eucal{A B C D E F G H I J K L M N O P Q R S T U V W X Y Z} \end{array} \] \end{document}   But this feels very inefficient. How to send data to a serial port and see any answer? On Linux, I want to send a command string (i.e. some data) to a serial port (containing control characters), and listen to the response (which also usually might contain control characters).   How can I do this as simplest as possible on Linux? An example is appreciated! I am trying to learn how to find the product of non-disjoint cycles, as you may have guessed from the title. I have the basic idea, but I do not understand it entirely.  I am trying to find $(1352)(256)$ and $(1634)(1352)$, multiplying from right to left.  Please give me some hints/suggestions.  Thank you. How do I find values not given in (interpolate in) statistical tables? Often people use programs to obtain p-values, but sometimes - for whatever reason - it may be necessary to obtain a critical value from a set of tables.   Given a statistical table with a limited number of significance levels, and a limited number of degrees of freedom, how do I obtain approximate critical values at other significance levels or degrees of freedom (such as with $t$, chi-square, or $F$ tables)?  That is, how do I find the values "in between" the values in a table? Triple Boot Lion+Win7+Linux A group such that $a^m b^m = b^m a^m$ and $a^n b^n = b^n a^n$ ($m$, $n$ coprime) is abelian? Let $(G,.)$ be a group and $m,n \in\mathbb Z$ such that $\gcd(m,n)=1$. Assume that $$ \forall a,b \in G, \,a^mb^m=b^ma^m,$$ $$\forall a,b \in G, \, a^nb^n=b^na^n.$$  Then how prove $G$ is an abelian group?  Some context: Some of these commutation relations often imply that $G$ is abelian, for example , or . This looks like another example of this phenomenon, but the same techniques do not apply. Power Series proofs $$\alpha(x) =\sum_{j=0}^\infty \frac{x^{3j}}{(3j)!}$$  $$\beta(x) = \sum_{j=0}^\infty \frac{x^{3j+2}}{(3j+2)!}$$  $$\gamma(x) = \sum_{j=0}^\infty \frac{x^{3j+1}}{(3j+1)!}$$  Show that $\alpha(x+y) = \alpha(x)α(y) + \beta(x)\gamma(y) + \beta(y)\gamma(x)$ for every $x, y \in\mathbb R$.  Show that $\alpha(x)^3 + \beta(x)^3 + \gamma(x)^3 − 3\alpha(x)\beta(x)\gamma(x) = 1$ for every $x\in\mathbb R$.  My work:  Ive noticed that $$\sinh(x)=\sum_{j=0}^\infty \frac{x^{2j+1}}{(2j+1)!}$$ and $$\cosh(x)=\sum_{j=0}^\infty \frac{x^{2j}}{(2j)!}$$ Can hyperbolic trig identies be used to solve these and if so how? got me wondering about the differences between these three ways of measuring size: a kibibyte, a kilobit, and the conventional kilobyte.  I understand that these measurements have different uses (data transfer rate is measured in bits/sec), but I'm not quite sure if I can tell the difference between Mb and MB and MiB.  Here is a comment, reproduced below, taken from  (emphasis mine).     The C64 has 65536 bytes of RAM. By   convention, memory size is specified   in kibiBytes, data transfer rates in   kilobits, and mass storage in   whatever-the-manufacturers-think-of-now-Bytes. Harddrives use T, G, M and k on the   label, Windows reports the size in Ti,   Gi, Mi and ki. And those 1.44MB   floppys? Those are neither 1.44MB nor   1.44MiB, they are 1.44 kilokibibytes. That's 1440kiB or 1'474'560 bytes. –   Third How to detect scene strips in the VSE on frame changes? I need to detect scene strips in the VSE on the current frame, meaning on what strip is the play head at this moment? However I can't seem to find the right type(?) to attach the ".strip_elem_from_frame" function to, in order to get the info/name/type on the strip. Here's a test script which must run in a screen with a sequencer with scene and movie strips in it:  import bpy  #update at frame change def currentStrip(*pArgs):     print("Current Strip: "+str(bpy.context.sequences[0].strip_elem_from_frame(bpy.context.scene.frame_current))) #Doesn't detect any strips at all.  #add frame change handle def attachAsHandler():     for f in bpy.app.handlers.frame_change_pre:         bpy.app.handlers.frame_change_pre.remove(f)     bpy.app.handlers.frame_change_pre.append(currentStrip)  currentStrip() attachAsHandler() Limit involving binomial coefficients: $\lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}}$ I am facing difficulty with the following limit. $$ \lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}} $$ I tried to take log both sides but I could not simplify the resulting expression. Please help in this regard. Thanks. What wand was used by Mr Graves at the time of Fantastic Beasts? We know that the movie Fantastic Beasts is set in 1926. From the movie, we learn that at this time, Dumbledore is a teacher at Hogwarts and Grindelwald is lose after escaping a previous arrest. In the movie we learn that      Mr Graves is in fact Grindelwald in disguise. Grindelwald is shown to be at least in his forties, maybe fifties so he should already be the owner of the wand since he stole it from Gregorovitch as a teen.   We know that Dumbledore won the wand from Grindelwald after defeating him in their historic duel. That duel is always described as being “the end of the dark wizard Grindelwald” so I’m assuming that it didn’t happen prior to the movie and that he didn’t escape after that duel, please correct me if you have cannon evidence stating the contrary.      In the movie, we see Graves’ wand a few time and it isn’t the (very recognizable) one used in DH as the Elder Wand. This could of course be a prop error but since David Yates was also the director on DH and a lot of the visuals are similar I’m assuming it wasn’t. Graves could of course be using another wand in the fear that someone could recognize it but he would still be the rightful master of the Elder Wand. At the end of the movie, Newt is the one that disarms Graves/Grindelwald.   In all logic, he should have been made the new owner of the wand, whether Grindelwald was actually using it or not. That would have made it impossible for Dumbledore to win it from Grindelwald later.  So my question is: What wand is that character using at that specific time? Is he already the owner of the Elder Wand?      And if so, why wasn’t Newt made the new owner of the wand? Panasonic Lumix G 25mm f/1.7 ASPH vs Olympus M. Zuiko 25mm f/1.8 Index with exercise numbers Are these plural or singular?