Let $a_n$ and $b_n$ be two positive sequences such that $$ b_n =\frac{a_1 + a_2 + ... + a_n} {n}  $$ Prove that $\lim_{n\to \infty} a_n = 0$ implies $\lim_{n\to \infty} b_n = 0$. Is the converse true?
Prove that if $\lim_{n \to \infty}z_{n}=A$ then:  $$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$ I was thinking spliting it in: $$(z_{1}+z_{2}+\cdots+z_{N-1})+(z_{N}+z_{N+1}+\cdots+z_{n})$$ where $N$ is value of $n$ for which $|A-z_{n}|&lt;\epsilon$ then taking the limit of this sum devided by $n$ , and noting that the second sum is as close as you wish to $nA$ while the first is as close as you wish to $0$. Not sure if this helps....
How to capture a still image from digital video? How to show $\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$? Show that $\,\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$.   I'm thinking right now (though not getting anywhere with it) that I want to expand out the summation portion to $i!/2!(i-2)!$ and simplify from there? Not sure if that will help, not to mention if I put $1$ in for $i$ I get $1/-2$ which I don't think is right.   Anyone care to shed some light on the subject?   Thanks. How can I make a surface look wet in cycles? I am trying to recreate a wet road with lights reflecting off of it. I started by making the object glossy but this did not give a very realistic look  I think I need to increase the specularity of the reflection to make it look more real but I can't find this option anywhere in cycles. Also what other techniques can I use to make a surface look wet in blender? I can link directly due to licening but  is what I am trying to recreate. Kali Linux: apt update returns "Hash Sum mismatch" error I'm experiencing exactly the same issue as described in this question: . Before you mark this as a duplicate however, I have tried the solutions posted there, as well as on numerous other sites, including:  sudo apt-get clean sudo rm -rf /var/lib/apt/lists/* sudo apt-get update   Editing /etc/apt/sources.list with alternate official mirrors, such as  deb http://mirrors.ocf.berkeley.edu/kali kali-rolling main non-free contrib or deb https://http.kali.org/kali kali-rolling main non-free contrib  Everything worked after I first imported the VM. I ran sudo apt update and it found some ~650 packages to upgrade. I ran sudo apt upgrade and it encountered an error partway through. That error was solved using sudo apt --fix-broken install, but that is when this hash sum error began. Unfortunately due to hours of troubleshooting I no longer have the details of the earlier error, but I believe it was an error extracting a package due to corrupt data.  I've tried multiple official mirrors, but I get the same error. Additionally, when I downloaded the Packages.gz file  on my Windows machine (VM host) and computed the SHA256 hash, I got the exact hash that apt printed as the expected value. This led me to believe that the error was not with the mirror but with my VM.  The next thing I tried was wget https://mirrors.ocf.berkeley.edu/kali/dists/kali-rolling/main/binary-amd64/Packages.gz followed by sha256sum Packages.gz, which provided yet another different hash output.  To be clear, I have seen 3 different hashes for the same file:   The "correct" hash shown by apt as expected, which is the one that windows also produced after downloading the file using a browser The incorrect hash calculated by apt, which led to the error A different hash calculated by sha256sum after downloading the file using wget using the same URL as for the browser download   I should also note that I have only been referencing the SHA256 hash in each step. The other hash functions are also mismatched when I run sudo apt update, but the file size is the same. I had considered that downloads might be failing due to limited disk space (it is a VM after all) but I don't think that's the case.  What am I missing? Understanding the differential $dx$ when doing $u$-substitution Some publications capitalize He, Him, etc., when referring to a god.    What is this practice called? 'Proper noun' is rather generic. Physics are not simulated past a certain frame? I've set up the physics of a swinging wrecking ball and set the animation to run for 240 frames.  This was not long enough for the entire "swing" of the ball-and-chain.   I'm unable to reset the timeline, but I can set the end frame to (say) 1000 frames, but when I try to play the animation, the animation progress halts at frame 240 and I cannot get it to go any further.   There are no markers or keyframes in the timeline.  What should I do to "force reset" the animation-timeline, to allow changes to the length? Textcase, Listings, and Tilde Consider the following MWE:  \documentclass{scrreprt}  \usepackage{textcase}  \RequirePackage{listings} \usepackage[american]{babel}  \begin{document}     \MakeTextLowercase{A~B} \end{document}   LaTeX fails with the error  ! Undefined control sequence. &lt;to be read again&gt; az                      b l.9     \MakeTextLowercase{A~B}  ?    The error seems to lie in the combination of MakeTextLowercase, the listings package and the tilde. Whenever I remove any of these three components, everything works fine. Does anyone have a clue where the problem is? Are there attacks that break collision resistance but not preimage resistance? Are there any examples of attacks on hash functions which:   break collision resistance and second preimage resistant, or break collision resistance and preimage resistant?   I have looked at Rogaway's paper1, but it seems complicated, hence was wondering if a succinct example exists.  1)  Phillip Rogaway and Tom Shrimpton. Fast Software Encryption (FSE) 2004, LNCS vol. 3017, pp. 371&ndash;388, Springer, 2004. Is it touch(1) from coreutils-8.4-37.el6.x86_64 or my brain which is broken?  $ touch abc $ LANG=C stat abc   File: `abc'   Size: 0               Blocks: 0          IO Block: 4096   regular empty file Device: fd04h/64772d    Inode: 10485773    Links: 1 Access: (0664/-rw-rw-r--)  Uid: (3060399/ nodakai)   Gid: (  418/   quant) Access: 2016-10-14 18:42:06.189751847 +0800 Modify: 2016-10-14 18:42:06.189751847 +0800 Change: 2016-10-14 18:42:06.189751847 +0800 $ touch -a abc $ LANG=C stat abc   File: `abc'   Size: 0               Blocks: 0          IO Block: 4096   regular empty file Device: fd04h/64772d    Inode: 10485773    Links: 1 Access: (0664/-rw-rw-r--)  Uid: (3060399/ nodakai)   Gid: (  418/   quant) Access: 2016-10-14 18:42:17.374235446 +0800 Modify: 2016-10-14 18:42:06.189751847 +0800 Change: 2016-10-14 18:42:17.374235446 +0800 $ touch --help | grep 'access time'   -a                     change only the access time   As you see, it was not only atime but also ctime which was updated by touch -a !?!?  The filesystem is ext4 over LVM if that makes any differences. Was it just coincidence that Tahir Mahmood went on stage? In episode 3 of “Bodyguard,” Tahir Mahmood steps onto stage at St. Mary’s college as the Home Secretary is speaking and the two of them, plus others, are blown up by a bomb. In the next episode we discover that the stage had a weight trigger and that his arrival set it off. This heavily implies that whoever asked him to go on stage was affiliated with those who planted the bomb.  We know that Mahmood was sent on stage by advisor Rob Macdonald as part of a scheme, together with other members of the home secretary’s staff, to embarrass her. The audience knows that some of the people in on this scheme have been plotting with the U.K. security services. This, together with interference in video recordings of the event, characteristic of the security services, implicates the security services in the assassination.  In the final episode it is revealed that the bomb had been set by members of an organized crime gang who had gotten the explosives from an Islamic terrorist. A senior member of the police force had been helping them by leaking information. So, were the security services involved? If so, why? If not, who got Mahmood to go up on the stage? Was it just coincidence that he went up there at the right time? He who must not be named? (aka Anyone can ping everyone in chat) For some or other reason, typing @ᔕᖺᘎᕊ in a chat room pings everyone who's been in that chat room for the last 7 days...    Seems to work in all chat rooms (I only tested 2 as I don't want to upset too many people...)  How should we proceed? When I post an answer to a brand new question I usually find that someone else has submitted that same answer just moments before. (, usually ;-) )  What I've started to do lately is (I think) the right thing.   If I notice that another answer predates mine, and that I add no value over it, I immediately delete my own answer, and upvote the other guy.  I recommend that you too do the same. Otherwise the message will be diluted.  Similarly, if after posting your "answer" you realise that you are in fact wrong (and you've just misread the question) -- you can actually delete your answer.  Is this reasonable?  (The reason why I'm writing this down is because it feels completely counter-intuitive to delete your own stuff. But it really saves everyone time and effort) double perpendicular symbol (for independence) I found from  that I could use the independence symbol:     \newcommand\independent{\protect\mathpalette{\protect\independenT}{\perp}} \def\independenT#1#2{\mathrel{\rlap{$#1#2$}\mkern2mu{#1#2}}}    I was wondering how this works. Can anybody explain? I don't know the commands mathpalette, rlap, and mkern2mu. Gnome-shell seems to like to leave one monitor as "fixed" when you switch workspaces.  Additionally, there is no preview in the Activities view of my secondary monitor.  Also, on the second monitor, where there should be either empty desktop space, or a copy of the panel, I just have a white bar.  How can I get gnome-shell to treat my secondary monitor like the primary one? Stack Overflow as a reference in a professional paper/presentation Suppose I'm doing an academic paper, journal article, or other professional presentation.  Would you say that it would be possible to use Stack Overflow as a reference?    Of course I'm pretty sure that the answer here would be "yes."   So here's the follow up: When, if ever, would it be a good idea to cite a Stack Overflow question as a reference in an professional paper/presentation of some kind? I want to convert .mov to .mp4! I saved the video from iMovie but it is in .mov format! How do I convert it to .mp4?  I'm using a MacBook and I want to share it to my phone, but my phone can't open .mov format video. So the vending machine tore my $5 in pieces. What now? The vending machine swallowed half of my $5 bill, and the other half remained in my hand. Can I get my $5 back, or is it all lost forever? Can you win a game of Splendor if you still have cards in reserve? Stopping Dwarf Fortress from auto-pausing Is it possible to prevent DF from pausing every time something comes up (liking hitting gold or whatever). The reason is that I am doing extensive mining and I want to leave the process unattended till it is over.