javascript/jQuery sort array
Sort Array Elements (string with numbers), natural sort
Why are these estimates to the German tank problem different? Show that $\sum_{n = 1}^\infty n^qx^n$ is absolutely convergent, and that $\lim_{n \rightarrow \infty}$ $n^qx^n = 0$ I'm having trouble with proving the following for my math study:  Let $x$ be a real number with $|x| &lt; 1$, and $q$ be a real number. Show that the series $\sum_{n = 1}^\infty n^qx^n$ is absolutely convergent, and that $\lim_{n \rightarrow \infty}$ $n^qx^n = 0$  I tried this:  $\sum_{n = 1}^\infty n^qx^n$ = $x + 2^q + 3^qx^3 + ...$ $= x(1 + 2^qx + 3^qx^2 + ...)$  So far, I don't know how I have to do this, I tried to apply the root test on this, but for that I would need to compute the lim sup of the series, so I thought that was not correct. Could you explain me how to prove this?  Thanks in advance! Best way to create an system of equations environment? Continuum between addition, multiplication and exponentiation? I noticed this  which attempts to find the shades of grey between a linear and log scale where results are between zero and one.  However, I was looking for the more general case where we find the continuum between the operators themselves - add, multiply, and exponentiation or even higher.  The function could be a hyperoperation$(n,x,y)$, where $x$ and $y$ are the numbers to operate on, and $n$ is allowed to be a non-integer number for the type of operator. So $n=1$ would be 'addition', $n=2$ would be multiplication, and $n=3$ would be exponentiation. But one would also have $n=2.5$ to be the shade of grey BETWEEN multiplication and exponentiation, or $n=3.5$ or $4$ for beyond exponentiation and tetration).  I want the formula to work with real numbers, not just integers. Any thoughts?  These related posts may also be of interest: Puppet client not connection to master - SSL connect Error I installed the puppet master and agent on the same machine. When the client is started, I got the following error message.  puppet agent --server=agent.com --no-daemonize --debug    err: Could not retrieve catalog from remote server: SSL_connect returned=1 errno=0 state=SSLv3 read server certificate B: certificate verify failed.  This is often because the time is out of sync on the server or client How to show that $\int_{0}^{\pi}{\sin^5 x\over 1+\cos^3 x}\mathrm dx=\ln 3?$ Given the integral $(1)$     $$\int_{0}^{\pi}{\sin^5 x\over 1+\cos^3 x}\mathrm dx=\ln 3\tag1$$   This integral seems to be quite tough, I can't even know where to start. I have try substitution $u=1+\cos^3 x$ and $u=\cos^3 x$, can seem to remove the x variables.  We have the following identities   $$\sin^5a={5\over 8}\sin a-{5\over 16}\sin 3a+{1\over 16}\sin 5a\tag2$$  $$\cos^3 a={3\over 4}\cos a+{1\over 4}\cos 3a\tag3$$  We may write $(1)$ as  $$\int_{0}^{\pi}{\sin x(1-\cos^2 x)^2\over 1+\cos x(1-\sin ^2 x)}\mathrm dx\tag4$$  $u=1-\sin^2 x\implies \mathrm du=-2\sin x\cos x\mathrm dx$ then $(4)$ becomes  $${1\over 2}\int_{0}^{1}{(1-u)^2\over \sqrt{u}+u^2}\mathrm du\tag5$$  $u=v^2\implies du=2vdv$ then $(5)$ becomes  $$\int_{0}^{1}{(1-v^2)^2\over v+v^4}\cdot{v\mathrm dv}\tag6$$ Simplify to  $$\int_{0}^{1}{(1-v)(1-v^2)\over 1-v+v^2}\mathrm dv\tag7$$  I guess using partial decomposition of fraction and integrate term by term, seem lengthy process  How else can we prove $(1)$? We should make books of our best content! And that's a good idea!  We have a lot of high quality content here, certainly enough for a small book.   I'm not saying that it necessarily has to be integrated into the main site, but this is just a suggestion if anyone wants to implement it.   Stack Exchange probably could make money selling the physical books though. CC-BY-SA allows selling, as long as you allow others to make copies of your book, and credit the author.  What Wikipedia does is you can custom order any set of articles as a book (it even has a nice set of authors at the end, to satisfy the BY part of CC-BY-SA).  This could even tie into the docs feature. How do I start applications automatically on login? How can I make an application automatically start when I have logged in? I've got a problem while unwrapping UVs. When I'm unwrapping UVs it showing me this message.     Why is this happening? How to delay a command block I have created a short command block stack, one block teleports the player and the other plays block.piston.extend. My problem is that it teleports the player before the player can hear the sound. I would like to know if there is a way to delay the command from inside that command block. I have looked at other solutions but they all involve some complex scoreboard timer thing. If you could give me a solution that doesn't use that system or maybe you could explain it better.  This is the command I need to delay:  /tp @p -586.634 64 -430.697   If anyone could help I would really appreciate it, so thanks in advance! show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$ How to render a background image in 2.8? I'm trying to "add" an 3D object onto a photo. I have added the photo to the scene in three ways so far, but none of them render (eevee):   Camera > Background Images Empty a. Add > Empty > Image (Empty gets added to current Collection) b. Add > Image > Backround (Empty gets added to Scene Collection)   These all show up in the 3D View and I was able to align the camera and the object so its perspective matches the scene in the photograph. But when I Render the scene, I only see the object, not the background image.  For the latter, it's simple: Empties are not rendered at all. There are few questions about this, since 2.8 has changed how it handles background images. The most popular seems this one (it includes references to others):      But I seem not to be able to find a method to have the background image render.  Another method might be to use the Compositor (Nodes) to Mix or Alpha Over an Image with the Render Layers. But this does not give the same alignment as achieved with the background image (also this requires that the World background does not get rendered (Alpha channel); for Eevee, this can be achieved by checking Film>Transparent).  What other options are there?  Docs: There is by now a captcha-problem Food in the Star Wars Universe? Rounding Percentages I know that there's a difference between relativistic and rest mass. Relativistic mass is "acquired" when an object is moving at speeds comparable to the speed of light. Rest mass is the inherent mass that something has regardless of the speed its moving at.  When fusion happens, a certain percentage of the rest mass of two hydrogen atoms is converted into energy. I understand that $E = m_{rest}c^2$.   How does the rest mass of an object turn into energy? Does this involve particle physics? Doesn't this suggest that rest mass can be viewed as some sort of potential energy? It seems to me that it's a sort of locked energy that has to undergo a certain process to expel its energy to surrounding particles.  Also, Why does only a small percentage of rest mass turn into energy? Why not all of it. What dictates how much rest mass gets converted into energy? Ifstream read function doesn't load into vector Is there any way to create multiple insert statements in a ms-access query? What does '>/dev/null 2>&1' mean in this article of crontab basics? Which equals operator (== vs ===) should be used in JavaScript comparisons?