Simple tuple in Python
Understanding slice notation
I want to find the interval of convergence of $$\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}(2x)^n$$ By the root test, $\sqrt[n]{|(\frac{n}{n+1})^{n^2}(2x)^n|}=|(\frac{n}{n+1})^{n}(2x)|$  And $(\frac{n}{n+1})^n=(1-\frac{1}{n+1})^n\to\frac1e$ as $n\to\infty$  So I've got an open interval(of convergence) s.t. $|\frac1e 2x|&lt;1\implies |x|&lt;\frac e2$.  Now I need to figure out whether the series is convergent or not at the end points, $x=\frac e2, -\frac e2$.  So there are two series I need to deal with, namely, $\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}e^n$ and $\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}(-1)^ne^n$.  Suppose $(\frac{n}{n+1})^{n^2}$ is increasing. Then $$(\frac{n}{n+1})^{n^2}&lt;(\frac{n+1}{n+2})^{(n+1)^2}\iff 1&lt;(\frac{n+1}{n+2})^{(n+1)^2}(\frac{n+1}{n})^{n^2}$$ $$\iff 1&lt;(1-\frac{1}{n+2})^{2n+1}(1+\frac{1}{n^2+2n})^{n^2}$$  But $(1-\frac{1}{n+2})^{2n+1}\to\frac{1}{e^2}$ and $(1+\frac{1}{n^2+2n})^{n^2}&lt;(1+\frac{1}{n^2})^{n^2}\to e$  So I have $1&lt;\frac1e&lt;1$, which is a contradiction. Hence $(\frac{n}{n+1})^{n^2}$ is not increasing. But I don't know if it's decreasing. (is it true that $(1+\frac{1}{n^2+2n})^{n^2}\to e$?)  If $(1+\frac{1}{n^2+2n})^{n^2}\to e$ is true, then $(\frac{n}{n+1})^{n^2}e^n$ converges to some positive number, so $\sum(\frac{n}{n+1})^{n^2}e^n$ diverges. But I don't know if it's decreasing: can't apply the alternating series convergence test(Leibniz's criterion).  So, can you help me with the two end points? Resolving a Git conflict with binary files I've been using Git on Windows (msysgit) to track changes for some design work I've been doing.  Today I've been working on a different PC (with remote repo brian) and I'm now trying to merge the edits done today back into my regular local version on my laptop.  On my laptop, I've used git pull brian master to pull the changes into my local version. Everything was fine apart from the main InDesign document - this shows as a conflict.  The version on the PC (brian) is the latest one that I want to keep but I don't know what commands tells the repo to use this one.   I tried directly copying the file across onto my laptop but this seems to break the whole merge process.  Can anyone point me in the right direction? How do I reset all progress in the game "just shapes and beats"? The game  has a very unique intro that I would like to show my friend again for the first time. I'm not sure how to reset all my progress on this game to achieve this though. Any thoughts? Converting Raw HTTP Request into HTTPWebRequest Object Local collection of Python packages: best way to import them? Is there a nice example of integral domain $R$ and $s,t \in R$, with $Rs \cap Rt$ not principal but lcm($s,t$) exists? I am currently searching for an integral domain $R$ which allows lcm, but where the intersection of two principal ideals is not necessarily a principal ideal (or at least an example with two certain elements).   As converse example (where the sum of two ideals is not principal but $R$ allows gcd) is $\mathbb Z[X]$. It allows gcd, but obviously $\left&lt;X+1,2\right&gt;$ is not principal.  An example of an integral domain where the intersection of two principal ideals is not principal would be $\mathbb Z[\sqrt{-5}]$ with the ideal $\left&lt;2\right&gt; \cap \left&lt;1+\sqrt{-5}\right&gt;$, but this doesn't allow lcm.  Since I can't find a nice example, I put the question here. Maybe someone has already seen this? (I am not completely sure, one even exists, but pretty certain there is an example. In the case my hunch is wrong, I would be happy, if someone could proof me wrong.) Error 404 : While using flag Recently on Stack Overflow I got privilege to flag a post. So when I try to flag a post as duplicate. The diaglog box is not properly loading.         However I am easily able to use other flag option like "other","spam" or "offensive".   "other"        "spam"        "offensive"        I am using Google Chrome Version 28.0.1500.71 and Win XP Operating System. I have tried in firefox as well but it is showing me same problem. Trigonometric Uncertainty Propagation I have brand new machine.  I want to install Ubuntu and one more Linux distributions on same machine. When the machine is up I want to have the option to choose which Linux to load.   Is it possible?  How to do it? I have interesting bug that I have experienced before but has gone away on its own. This one is not going away though.  I have a giant stripe or chunks missing from my desert biome, I would say probably 7 chunks long.  I'm guessing the chunks are corrupted or something, what was hoping to figure is whether it can repaired or reverted.  Here is a picture of the problem. I've just been downvoted. How should I react? Centrifugal force acting on ring Take for example the case of a rod rotating about an axis passing through its centre of mass and perpendicular to it. It has a ring hung from one of its sides. The rotation of the rod causes the ring to move out wards and ultimately fall off the rod.   I asked my teacher what causes the ring to move out wards and he said the centrifugal force acting in the frame of reference of the ring causes motion. What exactly is the ring reference frame? Is it a rotating frame? How would u visualize motion in this frame of reference? I am new to the netplan on Ubuntu 18.04, just discovered it exists today. I have an interface that I am trying to add to create a floating IP on google cloud. It is based on this example:   I tested the example, and it works, but now I try to do the same on ubuntu, and I don't know how to convert this:  cat &lt;&lt; EOF &gt;&gt; /etc/network/interfaces auto eth0:0 iface eth0:0 inet static     address 10.190.1.1     netmask 255.255.255.255 EOF   to netplan.  The output for ls /etc/netplan is '50-cloud-init.yaml'. and the output for cat /etc/netplan/*.yaml is:  network:     version: 2     ethernets:         ens4:             dhcp4: true             match:                 macaddress: 42:01:0a:8e:00:3e             set-name: ens4   my ifconfig:  # ifconfig  ens4 ens4: flags=4163&lt;UP,BROADCAST,RUNNING,MULTICAST&gt;  mtu 1460     inet 10.142.0.62  netmask 255.255.255.255  broadcast 0.0.0.0     inet6 fe80::4001:aff:fe8e:3e  prefixlen 64  scopeid 0x20&lt;link&gt;     ether 42:01:0a:8e:00:3e  txqueuelen 1000  (Ethernet)     RX packets 9430  bytes 1635180 (1.6 MB)     RX errors 0  dropped 0  overruns 0  frame 0     TX packets 13383  bytes 1513428 (1.5 MB)     TX errors 0  dropped 0 overruns 0  carrier 0  collisions 0   What is the equivalent of what he said to put in the file, for netplan? Using sed with an integer variable I want to sed using a variable.  For example, sed -n '1,10p' file. So, instead of 10, I want to use $k where k is an integer. Are there any non-decreasing securities on the stock market? Are there any securities traded on the stock market (i.e. ETPs) that never decrease in value?    As an aside (in case anyone mentions bonds):  I'm aware there's ETFs for AAA-rated government bonds, but even these dip significantly.  For example, the  has dipped by 9% in a 85-day period before (21/08/2016 to 13/11/2016). How do I get the form id of my form? I've just begun to make my modules and I'm at a standstill. I got this code from someone else and am trying to customize it:  function hook_form_alter(&amp;$form, &amp;$form_state, $form_id) {   if ($form_id == 'the form id for the node form') {     $form['#submit'][] = 'my_custom_submit_handler';   } } We consider the differential equation $y'=x+y^2$ and for any $\eta \geq 0$ we denote by $\phi_\eta:(\alpha_\eta,\beta_\eta)\rightarrow\mathbb{R}$ the unique saturated solution of the Cauchy Problem $y'=x+y^2$, $y(0)=\eta$. I am required to do the change of variable $z(x)=y(-x)$ and I succeded here, obtaining $y'(-x)=y^2(-x)-x$ and so $z'(x)=x-z^2(x)$, then I was required to represent the $0-isocline$ and it was also simple as I obtained a parabolla and also I was required to write the equation of the tangent in the point $(0,\eta)$ which was also pretty simple as I obtained $\phi'_\eta(0)=\eta^2$ and the equation of the tangent is $y=\eta+\eta^{2}x$. Now, I am required to find $\phi^{(5)}_0(0)$( that $5$ indicates the number of differentiations), which puzzles me as I don't know the form of the solution and if it can be found out and also I am told to show that $\frac{1}{\eta}\geq\beta_\eta$ What I did not prove after the first edit : and that for $\eta &gt; 0$, $\alpha_\eta=-\infty$ and for the Cauchy problem $z'=x-z^2,z(0)=\eta$ the saturated solution is defined up to $+\infty$. At this last ones I am lost. Any help, please? Edit: I have passed beyond finding $\phi^{(5)}_0(0)$ and I think I did show that $\frac{1}{\eta}\geq\beta_\eta$, now it remains to prove the last questions. Reversing the $T(n) = \frac{n(n+1)}{2}$ formula Can someone reverse this formula? Sorry if it's too basic, I'm an old guy and my math is rusty :-D  $s =\frac{n(n+1)}{2}$  If it ain't clear, by reverse I mean obtaining the value of n by providing s  Thanks I would like the user to have full rights on this folder (as well as all sub-directories and files in it):  ~/.blabla   currently owned by root.  I have found numerous posts (in this forum and elsewhere) on how to do this for files but I can't find a way to do it for whole folders. How do you start Windows 10 in safe mode? I've been through the Windows 10 upgrade, and am experiencing a recurring "INTERNAL_POWER_ERROR", which is stopping me from logging in.  I'd like to be able to try to boot to safe mode, but it appears that pressing F8 during boot is no longer enough, so how can I boot in safe mode (or any of the other recovery options) on a Windows 10 PC that doesn't let me log on?