Calculate the sum of infinite series with general term $\frac{n^2}{2^n}$.
Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$?
How to install the OS with a new Hard drive? Prove that $\phi(n) \geq \sqrt{n}/2$ So I'm trying to prove the following two inequalities: $$\frac{\sqrt{n}}{2} \leq \phi(n) \leq n.$$ The upper bound we get from simply noting that $\phi(n) = n \prod_{p | n}\left( 1 - \frac{1}{p}\right)$ and the fact that $(1 - \frac{1}{p}) \leq 1$. But how can we get the lower bound? I tried using the same expression for $\phi(n)$ but it seems to not really give me the inequality. Can you help? In bash, how do I escape an exclamation mark? I want to do something like bzr commit -m "It works!". I can sort of escape the exclamation mark by doing bzr commit -m "It works\!". However, then my commit message includes the backslash. How do I escape the exclamation mark, while still ignoring the backslash? In Avatar: The Last Airbender, Aang's eyes glow when he goes into the Avatar State. This seems to be the case with all Avatars, but Aang and the other airbending Avatars tattoos glow too.      Is there any reason for why airbending Avatars tattoos glow as well as their eyes? /boot full of kernels – what to delete? We have some Ubuntu 16.04 servers. unattended-upgrades are automatically enabled since 16.04 and /boot is a separate partition. Due to the automatic security updates, the boot partition is running out of space with new kernels. We can't reboot the systems (for availability reasons), so the machine is still using the penultimate kernel. Which kernels should I remove? All but the current, the oldest and the newest? Do you guys have some recommendations? I have also noticed that the newest kernel has the status &quot;Half Configured&quot;. This kernel would probably not work, so should I remove this one and use an older kernel? Output of dpkg -l | grep linux-image: ii  linux-image-4.4.0-21-generic --&gt; old kernel ii  linux-image-4.4.0-34-generic --&gt; current kernel ii  linux-image-4.4.0-36-generic --&gt; new kernel ii  linux-image-4.4.0-38-generic --&gt; new kernel ii  linux-image-4.4.0-42-generic --&gt; new kernel ii  linux-image-4.4.0-45-generic --&gt; new kernel ii  linux-image-4.4.0-47-generic --&gt; new kernel ii  linux-image-4.4.0-51-generic --&gt; new kernel ii  linux-image-4.4.0-53-generic --&gt; new kernel iF  linux-image-4.4.0-57-generic --&gt; new kernel I'm one of those exceptions that deviates from the norm: I run CyanogenMod without any Google apps.   Is it possible to download the Stack Exchange Android app .apk directly? I want to be able to have LinkedList.contains() return true for a custom comparator.  Suppose that I have 1 LinkedList and 2 objects  LinkedList&lt;MyObject&gt; myList = new LinkedList&lt;MyObject&gt;();  MyObject a = new MyObject("HELLO"); MyObject b = new MyObject("HELLO");   Technicaly, both objects are identical in terms of comparison (MyObject implements Comparable)   ( a == b ) == true  however, when I do the following, myList does not return true for myList.contains(b)  myList.add(a) myList.contains(b) // == false   I think its because contains will check object reference and see that a and b are 2 distinct objects. Is there any way I can make it so I don't have to extend LinkedList to compare those objects? Is Dexterity overpowered compared to Strength? How do I get my players to form a PC party without just forcing them to? I believe that it is generally a good thing that all the PCs travel together, as a "party". This way the GM has to manage one environment – not, say, five. The problem is how the GM can ensure such a relationship between the characters.  One simple and really great solution is to require the players to design their characters so that they know each other. This has the added advantage of cross-backstories – as the players try to adjust their stories to fit together, they get and create hooks for more narrative!  However, I am now experimenting with giving the players complete freedom at character creation and later meeting in-game. No luck so far.  It is not easy, as trusting someone with your life (what happens during combat) is not a trivial thing. Furthermore, people generally like others with the same behaviour/interests/skin colour. Fantasy is all about diversity – you do not character-generate a strong man with a sword, no, you generate a half-ogre with an axe never before seen!  How can the GM get players, with diverse characters who are strangers to each other, decide to have their PCs act as a coherent group. I absolutely want to avoid asking my players to just solve this through metagaming ("Hey Joe, think of a way to get together with those, I can't story-tell two separate groups at the same time!").  I'm not having the stereotypical problem where a bunch of PCs meet for the first time and are automatically so paranoid of strangers that they start trying to kill each other. My players have just created a bunch of characters who have their own legitimate interests, and are faithfully following those in completely different directions that don't result in a classic "party". I put the PCs in great peril together and everyone escaped by their own devices, but now I have to figure out get the guy who ran for the hills back together with the couple that entered the city. And, I want to do it by resorting only to reasonable in-game events, not out-of-game metagame suggestions. Radio antenna producing waves in the visible spectrum If a radio could produce waves in the visible light spectrum, what would the result be?  This is a thought experiment that I've pondered for a few years now.  I realize there are a few/many real-world constraints, but if we lifted these constraints for the sake of thought, what could we expect?  Personally, I don't see why we wouldn't observe visible light emitting from the antenna, disregarding any light from Blackbody Radiation. Are tile bonuses counted twice if 2 new words are formed? An example:    O  O?   There is a triple letter tile to the right of the bottom O, indicated with a ?.  If I place an X on it, making the words OX (vertical) and OX (horizontal), do I score 50 points (scoring the tile twice) or 34 (scoring it once)? HTML from advertisement not rendered correctly I have many times found CSS written on my screen besides images from Stack OverFlow.   It is associated with the advert from Rackspace  Here is a screenshot: Unable to mount NTFS external hard drive What would be the nicest proof of the following theorem:  If $\lim\limits_{n \rightarrow \infty} a_{n} = \infty$, then $\lim\limits_{n \rightarrow \infty} \left(1 + \frac{1}{a_{n}} \right) ^ {a_{n} } = e$. If $\lim\limits_{n \rightarrow \infty}b_{n} = 0$, then $\lim\limits_{n \rightarrow \infty} \left(1 + b_{n} \right) ^ {\frac {1} {b_{n}} } = e$.  I somehow failed to find a proof here on the website and in the literature. There are four options. Which of them is correct?- (a)${p\over k}$ (b)${k\over p}$ (c)$\log({p\over k})$ (d)${\log p\over k}$ I somehow want to use the rule for evaluating this kind of sum using integration, i.e. $\int_{0}^{1} f(x) \ dx=\lim_{n\to\infty} {1\over n}\sum_{r=1}^{n}f({r\over n})$  But the given expression $\lim_{n\to\infty} [{1\over kn}+{1\over k(n+1)}+{1\over k(n+2)}+\cdots+{1\over k(n+p-k)}]={1\over k}\lim_{n\to\infty}{1\over n}\sum_{r=0}^{p-k}\frac{1}{1+{r\over n}}$, sum running from $0$ to $p-k$ instead of $n$. So, how to evaluate this limit? Thanks for assistance in advance. Decorating Python class methods - how do I pass the instance to the decorator? how to use \input{file.tex} correctly? Suppose I have two latex files file1 and file2 with the same preamble and so on. Then I want to create a file called bigfile which just puts the two files together. This is not working for some reason. Therefore I first tried to include just file1, this is what I have done:  \documentclass .... %% same as file1 ... same preamble as file1 \begin{document} \input{file1} \end{document}   The result is that I not only get file1, but also the preamble and so on. This is my first experience with putting files together in 1 file. Kindly provide me a solution :) Make links posted by users open in a new window Most of the time when I follow a user's link, I just want to see their site, look it over, then return to answer their question (not leave SO completely and not come back). Could we make the links users post open in a new window please?  EDIT: I originally wanted this implemented because it's what some other sites did, but I now definitely prefer SO's way. If I want to open a new tab, I middle-click/control-click (or whatever); if not, I just click. I'm in control of what happens. Would never want to go back now. How can I move files into subdirectories by count? I have around 100,000 files in a directory. As you can imagine, that's not an ideal situation, and I'd like to split the directory up into subdirectories. But the filenames aren't such where I can put them into subdirectories based on their names or anything, so I'd like to create, say, 10 directories and put about 1000 files in each. Is there an easy bash command or loop that would do this? What is the difference between "till" and "until"? What is the difference between till and until?    When to use till or until? Please explain with examples.