If $x=(x_1.x_2,x_3,...,x_n,...)$ and $\sum^\infty_{k=1} |x_k|^p&lt;\infty$,$p\geq 1$ can you help me to show that $\lim_{k\to\infty}|x_k|^p=0$?
Following the guidelines suggested in , I am going to post a proposed proof as an answer to the theorem below. I believe the proof works, but would appreciate any needed corrections.  Theorem If a series $\sum_{n=1}^{\infty}a_n$ of real numbers converges then $\lim_{n \to \infty}a_n = 0$
Kant introduces the notion of the noumenon in his critical philosophy; he also later demonstrates in his system that solipsism isn't possible.   Is the introduction of the noumenon essential to this demonstration, and thus justifies it's introduction? How to deal with high correlation among predictors in multiple regression? Suppose we have a situation like      A ball of some mass $m$ with some velocity collides with rod hinged at point $A$. Is momentum conserved in this situation? I know that hinge will give impulsive force on rod but that is internal force when we take ball and rod together as a system so that shouldn't stop us using conservation of momentum equation I saw this bug mentioned in image below, where in I was about to mark the question as duplicate..    Has you can see there is no radio buttons to select and go further.. And I am stuck..  Is this a KNOWN bug..  I am using Mozilla Firefox Browser..  Edit:-  As guided by @AzizShaikh, This kind of behaviour is observed when you have already flagged a post. In my opinion, I think the page as I mentioned above should not have come in first place, I mean don't allow the user to enter it doesn't belong here, or it is a duplicate section if his vote has been posted successfully for that section rather than showing go nowhere Im lost Screen ..  Can we do something about this...?? It seems that most people who use the ignored tags feature end up with a fairly long list of those tags.  Although , it is sometimes necessary to add new ignored tags manually (typically when adding some-tag*).  It would be a lot easier if the add box was above the long list of ignored tags rather than below. I suppose the same would apply to the favorites tags for consistency's sake. Ambiguity of “not.... because” You are not hired because you are qualified.   I think this means you are hired but it is not because you are qualified. But I think this can  also means you are not hired, and the reason for it is that you are qualified.   Does this ambiguity really exist or do I mistake anything? Let $f$ be defined on $\mathbb{R}$ by $f(x) = e^{-1/x^2}$ for $x$ not equal to $0$. and $f(0)= 0$. Prove that $f^{(n)}(0)=0$ for all $n = 1, 2,3$ ...  Do I need to use Taylor expansion from calculus class?  Any hint would be appreciated. Where is the flaw in this "proof" that 1=2? (Derivative of repeated addition) Consider the following:   $1 = 1^2$ $2 + 2 = 2^2$ $3 + 3 + 3 = 3^2$   Therefore,   $\underbrace{x + x + x + \ldots + x}_{x \textrm{ times}}= x^2$   Take the derivative of lhs and rhs and we get:   $\underbrace{1 + 1 + 1 + \ldots + 1}_{x \textrm{ times}} = 2x$   Which simplifies to:   $x = 2x$   and hence    $1 = 2$.     Clearly something is wrong but I am unable pinpoint my mistake. I've just bought my first Mac, a Macbook pro(with an hdd), and I'm thinking in buying an ssd and replace the hdd. But the computer came with no dvd of the OS. I know that in the hdd there is a restore partition, but if I remove the hdd, how can I install the OS? Converting mobi files to pdf files I have a Hanlin V3+ e-book, unfortunately it crashes too often when reading mobi files, so I want to convert them first into pdf, and then read pdf on V3+. Is there any tool for converting mobi files to pdf files? Thanks. When should I use  or >()? I know that &lt;, &lt;(), &lt;&lt; all are used to give input and  &gt;, &gt;() which are used to redirect output.  But I don't know what is clear difference between them and when to use them.  Please explain in detail and along with references if possible. Thanks Stack Overflow 'review' button is slow, and test questions are slower In the last few weeks, the 'review' button at the top of stackoverflow.com has become very slow. I can count to 20 in some cases (but not always).   In a perhaps related phenomenon, I believe that test questions in the queue often trigger a level of extreme slowness all their very own. If I click on a queue, or 'next' after a review, and the periods sit there throbbing for a good long time, chances are that what will show up will be a review audit. Can I replace my road bike's front wheel quick release with any quick release? I lost my road bike's quick release. Pictured are what I lost. Do I need to find a specific quick release that fits, or does one size fit all/most? Epson LX-310 not working after Cumulative Update 2017-11 Windows 10 I have Windows 10 Pro x32 and an Epson LX-310 printer that was working. Yesterday, Windows 10 automatically updated with 2017-11 Cumulative Update for Windows 10.   Today, I cannot print. The printer status is on. When I click "Print a test page", it quickly says the test page has been sent but the printer does not print it.   My printer is connected to a USB port. What wrong with my Windows 10? Are there any coding style guidelines for LaTeX? Why can my keyboard only digest 6 keypresses at a time? When I slam my hand onto my keyboard, only a maximum of 6 keys will be registered, I've tested this for the past few minutes and was unable to get more than 6. Is this the same on all keyboards? If not, what would be the distinguishing factor, If yes: why 6 specifically?  For clarification: I do not have a usecase for this, I am simply curious This is a question you can read everywhere on the web with various answers:  $ext = end(explode('.', $filename)); $ext = substr(strrchr($filename, '.'), 1); $ext = substr($filename, strrpos($filename, '.') + 1); $ext = preg_replace('/^.*\.([^.]+)$/D', '$1', $filename);  $exts = split("[/\\.]", $filename); $n    = count($exts)-1; $ext  = $exts[$n];   etc.  However, there is always "the best way" and it should be on Stack Overflow. What is the best way to find out a researcher's current affiliation? Let's say I know a name and a field, but not much more.   A google search seems like the most obvious approach, but unfortunately university personnel or research group pages often seem to be quite poorly maintained, and it is not uncommon that someone is listed even if they left more than a year ago.  Early-career researchers are particularly difficult to track down since they often have less of a "presence" on institution webpages and are likely to be moving relatively frequently. I made a script that should notify me when there's a new chapter of manga that I'm reading. I used the command notify-send to do this. The program works when I am trying to run it in terminal. The notification is showing. However, when I placed this in my crontab, the notification doesn't show. I'm pretty sure that the program is running since I made it to create a file for me. The file was created, but the notification didn't show.  Here's my script  #!/bin/bash    #One Piece Manga reminder     #I created a file named .newop that contains the latest chapter.     let new=$(cat ~/.newop)     wget --read-timeout=30 -t20 -O .opreminder.txt http://www.mangareader.net/103/one-piece.html  if (( $(cat .opreminder.txt | grep "One Piece $new" | wc -l) &gt;=1 ))     then         (( new+=1 ))         echo $new         echo $new &gt; ~/.newop         notify-send "A new chapter of One Piece was released."     else         notify-send "No new chapter for One Piece."         notify-send "The latest chapter is still $new."     fi         exit   And here's what I wrote in my crontab  0,15,30,45 12-23 * * 3   /home/jchester/bin/opreminder.sh Fix terminal after displaying a binary file