I found a proof that $\lim \sqrt[n]{1^2+2^2+...+n^2}=1$ by $\varepsilon$-language, but I think it's quite complicated and not sure that it's correct. My question is: 1- Is my proof correct? 2- Is there another simpler proof in the sense of $\varepsilon$-language? Please help me. Thanks. Solution: Let $a_n=\sqrt[n]{1^2+2^2+...+n^2}-1$. Then $a_n&gt;0$ and: $$ (a_n+1)^n= 1^2+2^2+...+n^2 = \frac{n(n+1)(2n+1)}{6}$$ By the binomial theorem we have: $ (a_n+1)^n = 1 + na_n+\frac{n(n-1)}{2}a_n^2+\frac{n(n-1)(n-2)}{6}a_n^3+\frac{n(n-1)(n-2)(n-3)}{24}a_n^4+\cdots+a_n^n $ Since $a_n&gt;0$, then $(a_n+1)^n&gt;\dfrac{n(n-1)(n-2)(n-3)}{24}a_n^4$ and therefore: $$\dfrac{n(n+1)(2n+1)}{6}&gt;\dfrac{n(n-1)(n-2)(n-3)}{24}a_n^4$$ This is equivalent to $$(n+1)(2n+1)&gt;\dfrac{(n-1)(n-2)(n-3)}{4}a_n^4 \Leftrightarrow a_n^4&lt;\dfrac{4(n+1)(2n+1)}{(n-1)(n-2)(n-3)}$$ $$\Rightarrow a_n^4&lt; \dfrac{8(n+1)^2}{(n-2)^2(n-3)}=\dfrac{8}{n-3}\Big(1+\dfrac{3}{n-2}\Big)^2&lt;\dfrac{128}{n-3}.$$ Thus: $a_n&lt;\sqrt[4]{\dfrac{128}{n-3}}$. For every $\varepsilon &gt;0$, take $N&gt;3+\dfrac{128}{\varepsilon^4}$, then for all $n\ge N$, $$ n-3 \ge N -3 &gt; \dfrac{128}{\varepsilon^4} $$ Thus: $ \varepsilon^4 &gt; \dfrac{128}{n-3} \Rightarrow \varepsilon &gt; \sqrt[4]{\dfrac{128}{n-3}} &gt;a_n $. Hence, $\lim a_n=0$ or equivalently, $\lim \sqrt[n]{1^2+2^2+...+n^2} =1$.
Determine value of the limit:  $$L=\lim_{n\to \infty}\sqrt[n]{1^2+2^2+\cdots+n^2}$$  My try:  $$1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$$  Hence: $$L=\lim_{n\to \infty} \sqrt[n]{\frac{n(n+1)(2n+1)}{6}}=?$$  But, come here, i do not know how, because please help me.
Pure virtual destructor in C++ On Mac, are there keyboard shortcuts to create a new Space? Or at least any way to automate this process with keyboard shortcuts? I want to not have to use the mouse to create Spaces. I want to use the keyboard for that. How to create a new partition from the existing partition? I wanna create a new partition from the existing root partition.My root partition is about 250GB.I wanna create a new partition say 150 GB taking space from this root partition.Can anyone help me with the procedure on doing this?  output: fdisk -l  karthick@karthick:~$ sudo fdisk -l [sudo] password for karthick:   Disk /dev/sda: 250.0 GB, 250000000000 bytes 255 heads, 63 sectors/track, 30394 cylinders Units = cylinders of 16065 * 512 = 8225280 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk identifier: 0x28219744  Device Boot      Start         End      Blocks   Id  System /dev/sda1   *           1          61      487424   83  Linux Partition 1 does not end on cylinder boundary. /dev/sda2              61       30395   243650561    5  Extended /dev/sda5              61         124      498688   82  Linux swap / Solaris /dev/sda6             124         186      498688   82  Linux swap / Solaris /dev/sda7             186       30395   242651136   83  Linux What does "Change" represent in the User Leagues? What does the Change field represent on the  for SO? Refer to the highlighted areas in the image below:   Rank: the user's ranking for the selected time period (week, month, quarter, year) Change: +/- change representing... ??? Total Reputation: the user's total reputation [Time Period] Reputation: the user's reputation for the selected time period (week, month, quarter, year)  So what is the Change field measuring? Calculating the average of variably sized blocks of numbers within a column? Prove that for every $n\geq m \geq1$ natural numbers, the following number is an integer:  $${\gcd(n,m)\over n}\cdot{n\choose m}$$  Where $\gcd$ is the greatest common divisor.  I tried to make it simpler by cancelling the $n$ from the left side, and making it $(n-1)!$ on the right: $\gcd(n, m) \cdot \frac{(n-1)!}{m!(n-m)!}$, but can't really go further.   This was problem B-2 on the . Could I use a 1/2TB USB 2/3 external HDD with an XBox 360? I'm considering getting an XBox 360 (plus Kinect). I need storage for other devices as well (movies, PC games, whatever an Ouya might need), so I'm considering getting a big external HDD. Can I store downloaded games (e.g. Fez) on such a device? I use OS X Mavericks and have some company storage drives mounted (using smb or cfs). (smb://username@address.path.toserver.com/directory/data). When I work with files on the drive or copy files to the drive, some hidden files are created. Most of these files start with dot-dash ._ and then the filename of the original file, like:  ._test001.png ._test002.png ._test003.png   And most of the times, the famous ._.DS_Store and .DS_Store are also there.  On my Mac, these . dot items are invisible, but my windows colleagues do see these files. How can I prevent OS X from making these strange invisible files?  On  (and other internet sources) you can find a 'defaults write' to disable the creation of .DS_Store files on your network drives. But this does not work in Mavericks:  defaults write com.apple.desktopservices DSDontWriteNetworkStores -bool true   For now I use this oneliner, but doing a rm -rf does feel wrong in this case.  find . -type f -name "._*" -exec rm "{}" \;   Edit: I don't want to delete the files or exclude them from copying. I want them not to be created in the first place.  Edit: I prefer a 'native' OS X solution over third party apps.  Edit: This questions has two answers, one for the ._ files and one for the .DS_Store files. The bounty is rewarded for dot_clean, this native command line app cleans all ._ files. Unfortunately is does not clear the .DS_Store files, this can effectively be done with Asepsis. Asepsis stores .DS_Stores in a fixed location, and not in the folder itself. There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$, what I tried is if possible then define $g(z)=f(z)-{1\over 2^{1\over z}}$ Then $g({1\over n})=0$ and so $g(z)$ is entire and its $0$ set has limit point in it and so $f(z)={1\over 2^{1\over z}}$ which is not analytic at $0$? Please help!  Edit: OOps! the way I defined $g(z)$ that is not entire! could any one give me hint? Why does Jack let Walt go? In Breaking Bad S5 E14, Jack not only lets Walt go, but also leaves a barrel of money to him after killing Hank. I don't understand why he would do this. Jack says that, with 7 barrels of money, he doesn't need to cook meth any more, so there is no reason to keep Walt (the witness of the killing) alive, let alone to give him so much money. What's the rationale behind this? Or it's just mercy or respect? How do I execute a command-line program from C# and get back the STD OUT results? Specifically, I want to execute DIFF on two files that are programmatically selected and write the results to a text box. Short story with competitors complaining to government about a new very popular synthetic food Synthetic foods are all the rage, and while some are new and novel, many are attempts to recreate the taste and texture of a natural foods, with varying degrees of success. One company introduces a new synthetic food that becomes wildly popular. The competition analyzes it, and based on discovering what natural creature it is based on, they petition the government to ban it.      They discover that the product is synthetic human flesh, so consuming it is not technically cannibalism, but it is similar, as they explain to the panel of government officials. Quantifiers "most" vs. "most of" This is a  about DNS/Hostnames resolution to IPs/Ports   Example 1  I'm running a web server on port 80 and another on port 87. I would like to use DNS so that www.example.com goes to port 87. How can I accomplish this using DNS only?  Example 2  I'm running a service on my server on a non-standard port. How can I get clients to connect to this non-standard port automatically? Can I use DNS? Is there some application specific support where DNS could indicate the IP and Port?   Example 3  Do some application protocols specifically support hostname awareness, and allow special actions to be taken based on this information? Are there other questions on Server Fault that cover some of these?  Commandeering: This question was originally asking about running IIS and Apache on the same server, but the same concepts can be applied to any server software receiving connections from clients. The Answers below describe the technical problems and solutions of using DNS and application protocol support to assign a port number for a client to connect. Removing columns in a SpatialPolygonsDataFrame in R? My spatial polygon data frame (SPDF) contains too many columns (variables) and I want to remove most of the columns entirely.   I know how to do this with a regular data frame in R, but I am unsure how to do this when dealing with object of class SpatialPolygonsDataFrame? What software is used to generate the reputation graphs on Stack Exchange? I'm particularly interested in the zooming functionality which is done very well. Were these graphs generated using off the shelf software or were they coded from scratch? I am crawling some data using Apache Nutch and storing it on my USB Drive. The OS itself is Ubuntu Live Boot OS. But for every restart or shutdown I am losing my data. I also tried suspending my Ubuntu user still data was not visible when I reinserted my live USB.  Is there any way I can keep the data intact and does not loose and application settings installations when I do restart Ubuntu.  Used USB installer provided by pendrivelinux.com at Expectation of random variables ratio What are Unity's keyboard and mouse shortcuts? This has been a old live-action movie from Asia, most likely Japan. I've seen it some time in the late 80's and I think it's been black/white only.  Earth is invaded by UFOs (small and round, like flying balls), coming from Mars.  Japan (I think) built some kind of super spaceship, initially hidden within some secret port in some kind of island/mountain.  The spaceship looks similar to a classic battleship, but with an added drill below the front.  In the movie's finale it's revealed that this is some kind of   super weapon/bomb that has to be manually piloted by someone. When launched it starts rotating and flies (formed like a bullet but with the drill/corkscrew head).   To defend Earth, the ship is sent on a mission to Venus, where the UFOs seem to originate. I don't remember the specific reason or conclusion here.  The ship starts underwater or launches from the mountain on the island, then taking off and fighting UFOs while leaving Earth's atmosphere.