I had a question regarding a proof for my analysis class  $$\lim\limits_{n \to \infty}\sqrt[n]{n^2 + 1}$$  I've tried a handful of things, but in all cases, I have either reached a point that has no logical connection to the result, or I don't know how to prove a step.  I'm not asking for the answer, but advice on how I should proceed with this problem would be helpful.
I have to prove that $$\lim_{n\to\infty} \left(1+n^2\right)^{\frac1n}$$ converges and give its limit. What I did was the following: $$\lim_{n\to\infty} \left(1+n^2\right)^\frac{1}{n} = \lim_{n\to\infty}(n^2)^{\frac1n}\lim_{n\to\infty}\left(\frac1{n^2} + 1\right)^\frac1n= \lim_{n\to\infty}(n^2)^{\frac1n}$$ Now that is where I am stuck - how to I continue with $\lim_{n\rightarrow\infty}(n^2)^{\frac{1}{n}}$? I really don't have any ideas left... Thank you very much! FunkyPeanut
Why didn't Dumbledore prevent the prosecution of Sirius Black? Dumbledore performed the Fidelius charm on the Potters, so he knew who the Secret-Keeper was1. That implies he knew not Sirius but Wormtail betrayed the Potters. Why didn't he prevent Sirius's conviction and suspect the explosion killing twelve Muggles was caused by Wormtail (and that Sirius cornered Wormtail instead of the other way around)?   1: Though in  it is said Dumbledore didn't know. I find it hard to believe the person casting the Fidelius charm doesn't know who the Secret-Keeper is. Which of $\{J_0,J_1,J_2\}$ and $\{J_0,J_+,J_-\}$ is a generator for the Lie algebra $\mathfrak{su}(2)$? I have just started learning about Lie algebra in the context of quantum mechanics and got confused with this: Some sources say the generators are $J_0,J_1$ and $J_2$ and some use $J_0,J_+$ and $J_−$. Which set is correct? Or if both are correct what key concept am I missing here? My understanding is that if have certain commutation relations then we know that Lie Algebra is such and such. But if we have two such choices then this understanding falls apart? How do I figure out what is $\mathfrak{su}(2)$ Lie Algebra in general? I have two users, user1 and user2, that are both members of groupA. user2 has a folder in their home directory called folderA. If they wish to allow read-write-execute permissions for all members of groupA, how would they do this?  What if folderA contains many files and additional folders that also need to have read-write-execute permission?  Information regarding groups is a little 'spotty' across the web, so I am putting my question here in the hope someone posts a clear answer that might help others out too.  Thanks! "Seen" status is earlier than the user's last post I posted what I think is a clever answer to:    But then I noticed the question was posted a while ago on Mar 2.  I wanted to see what the chance of the user logging back in and seeing my answer, but the  says the user was seen on Feb 19.  I thought this was supposed to be the date they last logged in? How could they have asked a question on Mar 2 if they haven't logged in since February?  Thanks Greatest common divisor of two positive integers Filter log file entries based on date range Is it possible to choose which OS will be loaded using a hardware switch? My keyboard has no "media" keys; can I control media without them? ID for Contact portion of Person Account Is it possible to get the ID of a Contact which is part of a Person Account?  I have a custom object which has a custom field referencing the Person Account, though it seems to be referencing the Contact directly. I have created a button which I am using to pass the ID into a Visualforce Page.  With the Insert Field drop-down on the create/edit button page I can put in {!Account.Id} or {!Contact.Id} but neither of these work. The Account ID is placed in the URL but is not the correct ID and the Contact ID is blank, which I assume is because the button is on a Person Account page and not a Contact Page. How is the derivative of $\textrm{Trace}\left\{ X^T A X B\right\}$ with respect to $X$ equal to $AXB + A^TXB^T$? Is it possible to include the tag-wiki associations in the Data Explorer? To do this would require two additional columns in the postTags table that associate a tag wiki excerpt and a tag wiki to their tag.  The tag wiki data is already available in the posts table. But there is no way to associate this data with its tags. The SEDE schema lacks the required columns to link the tag excerpt and the tag wiki to the tags table.  Additionally, the postTypes table only lists three posts types. It makes no mention of tag-wiki and tag excerpt posts. Perhaps the postypes table can be updated to reflect the additional post types? I have the following code:  MemoryStream foo(){     MemoryStream ms = new MemoryStream();     // write stuff to ms     return ms; }  void bar(){     MemoryStream ms2 = foo();     // do stuff with ms2     return; }   Is there any chance that the MemoryStream that I've allocated will somehow fail to be disposed of later?    I've got a peer review insisting that I manually close this, and I can't find the information to tell if he has a valid point or not. Formal Proof that area of a rectangle is $ab$ I tried to prove that the area of a rectangle is $ab$ given side lengths $a$ and $b$.  The best I can do is the assume the area of a $1\times1$ square is $1$. Then not the number of $1\times1$ squares that fit in an $a\times b$ rectangle is $ab$. Therefore area is $a\cdot b$. This does not seem rigorous however. Are there fewer positive integers than all integers? In our 6th grade math class we got introduced to the concept of integers. With all the talk about positive and negative, it got me wondering. Is the amount of elements in $\mathbb{Z^+}$ less than the amount of elements in $\mathbb{Z}$?  Here is how I thought of it. If we have $\mathbb{Z^+}$ and add one element to the "back" of it ($\mathbb{Z}^{\geq-1}$) there are certainly more elements in that new set, so there must be more elements in $\mathbb{Z}$ than in $\mathbb{Z}^+$ but on the other side if we try to express the amount of elements in them "numerically" (In a loose sense of the word) they both have $\infty$ elements.    So what is the right answer? Is there even one?  (Please note I am a bit of a layman when it comes to mathematics)  Edit: About the possible duplicate, I am not looking for a bijection between the two sets, I am looking for if they even have the same number of elements and why a bijection shows that they do/don't I want to know, if this bug is fixed, because all previous versions of Ubuntu didn't work for me. So I just want to know, should i leave Windows 10 and go for Ubuntu 17.10. Thanks! Is it possible to save frequently used expressions in QGIS? Is there any way to save frequently used expressions in the selection or field calculator dialogs? Obviously I can cut and paste them into a file where I might save them, but I mean within the context of QGIS like you can do in ArcMap. Converting a .lasd (Lidar) coordinate system? How can one change a coordinate system for a .lasd (Lidar) dataset?  The file is currently in UTM_Zone_15N and I am trying to convert it to IL_East. Printer not recognized on Ubuntu Libre Office I am running 14.04 and have a HP Deskjet 2545 wireless printer which is supported by Ubuntu. I have run HpLIP, the latest version and installed the printer.  The printer is recognised by Ubuntu, printer and network, but LibreOffice do not see the printer. Only a Generic printer.  I don't know what to do anymore and would like to get some good advice Matching a sentence with grep Why can't I update my Nexus 4 to Lollipop? As I understand it, Android Lollipop (5.0) is now generally available over-the-air. My Nexus 4 briefly offered for an hour or two to install it. I said yes, and watched it start to download. When I next went to unlock my phone, it obviously wasn't installed. Now, when I go into the Settings app, and manually check for updates, it says my phone is up-to-date. It's still on Android 4.4.4.  I don't particularly want to install the developer image, as that'll wipe all the data from my phone. Is there any way I can encourage it to update?  I'm in the UK, if it's relevant.