Proof for the limit of the following convergent sequence
On Cesàro convergence: If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $
Unary intersection of the empty set Similar to , what's the best way to stay up to date with the latest stable releases of Thunderbird? Why can't I find any manpages? I'm working on a Power8 server running Ubuntu 16.04.2 LTS:  $ uname -a     Linux power 4.4.0-75-generic #96-Ubuntu SMP Thu Apr 20 09:55:30 UTC 2017 ppc64le ppc64le ppc64le GNU/Linux  $ cat /etc/lsb-release DISTRIB_ID=Ubuntu DISTRIB_RELEASE=16.04 DISTRIB_CODENAME=xenial DISTRIB_DESCRIPTION="Ubuntu 16.04.2 LTS"   None of the usual manpages appear to be installed. Even a simple command like:  man man   returns   No manual entry for man See 'man 7 undocumented' for help when manual pages are not available.   This goes for a variety of other tools, like ls, grep, etc. and library functions like usleep, printf, and so on.  I've checked that at least some manpages are actually installed:  $ dpkg -l | grep -i manpages ii  manpages                           4.04-2                                     all          Manual pages about using a GNU/Linux system ii  manpages-dev                       4.04-2                                     all          Manual pages about using GNU/Linux for development ii  manpages-posix                     2013a-1                                    all          Manual pages about using POSIX system ii  manpages-posix-dev                 2013a-1                                    all          Manual pages about using a POSIX system for development   Am I missing something? Is this specific to the ppc64le architecture? Or is there something else I can/should install to access these manpages?  Update: As requested, I ran sudo mandb. The result was:  0 man subdirectories contained newer manual pages. 0 manual pages were added. 0 stray cats were added. 2 old database entries were purged.   There were no changes to man man and other such commands.  Additionally, the output of manpath was:  $ manpath -g /usr/man:/usr/share/man:/usr/local/man:/usr/local/share/man:/usr/X11R6/man:/opt/man $ manpath -c /var/cache/man/oldlocal:/var/cache/man/local:/var/cache/man Overriding the java equals() method - not working? The differential equation $$ {d^2\alpha \over dt^2} + {g \over L} \cdot \alpha = 0 $$  describes a 1-dimensional mathematical pendulum, where $\alpha $ is the angle, $ g = 9.82 $, and $ L = 0.2 $ is the lentgh of the string. What is the position of the pendulum after 1 second if the velocity at time equals zero is $ 0 $ m/s and the angle at time equals zero is $ {\pi \over 60} $ radians.  To my understanding this is a homogenous equation, ergo  $$ y'' + Cy = 0$$  And with the help of the characteristic equation I get the complex roots $ Ci \, $ and $ -Ci $.  This is a farily simple differential equation, but I have a feeling my calculations so far are wrong (mainly because I didn't expect to see complex roots for this equation). How does ssh run a command? I'm using Bash on both client and server. When running a command over SSH:   ssh &lt;host&gt; 'declare' gives a list of shell variables. ssh &lt;host&gt; 'mount' gives a list of mountpoints.   However, declare is a Bash builtin, while mount is an external command. How does SSH know which to run if there is a shell builtin and an external command with the same name on the server? How do malloc() and free() work? What are some examples of manifolds that do not have boundaries and are not boundaries of higher dimensional manifolds?  Is any $n$-dimensional closed manifold a boundary of some $(n+1)$-dimensional manifold? Finding what methods a Python object has Call to undefined function session_register() I'm working on a Ubuntu system, and my client has completely forgotten his administrative password. He doesn't even remember entering one; however it is there.  I've tried the suggestions on the website, and I have been unsuccessful in deleting the password so that I can download applets required for running some files. Is there a solution? What point (pt) font size are \Large etc.? In the preamble I have:  \documentclass[a4paper,11pt]{article} \usepackage{fontspec} \setmainfont{Arial}   What can be inferred about the real font pt size for the following?  \tiny, \scriptsize, \footnotesize, \small, \normalsize, \large, \Large, \LARGE, \huge, \Huge extract lines that match a list of words in another file I have file 1 which have those lines:  ATM 1434.972183 BMPR2 10762.78192 BMPR2 10762.78192 BMPR2 1469.14535 BMPR2 1469.14535 BMPR2 1738.479639 BMS1 4907.841667 BMS1 4907.841667 BMS1 880.4532628 BMS1 880.4532628 BMS1P17 1249.75 BMS1P17 1249.75 BMS1P17 1606.821429 BMS1P17 1606.821429 BMS1P17 1666.333333 BMS1P17 1666.333333 BMS1P17 2108.460317 BMS1P17 2108   And file 2 have a list of words:  ATM BMS1   So, the output will be like this:  ATM 1434.972183 BMS1 4907.841667 BMS1 4907.841667 BMS1 880.4532628 BMS1 880.4532628   I know it's really a duplicate question, but I tried all types of grep and sed and awk, maybe it will works with you guys with this tiny example  but I have a very huge file &gt; 1M lines and all previous way doesn't help  it return part of the lines that containing those words although there are other words in file 2 that matches the lines from file 1 Remove "Category:", "Tag:", "Author:" from the_archive_title I have the following code in my theme's archive.php:  &lt;?php the_archive_title( '&lt;h1 class="page-title"&gt;', '&lt;/h1&gt;' ); ?&gt;   This gives me titles like "Category: Russia", "Tag: America", "Author: John".  I would like to remove the "Category:", "Tag:" and "Author:" part and just display the category, tag and author names.  Does anyone know how to accomplish this?  Thank you. Evalute $ \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)} $ Evaluate $\displaystyle \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)}$.   $\bf{My\; Try::}$ Although we can solve it by converting into definite Integration.  But I want to solve it without Using Integration.  So $\displaystyle \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)} = \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{(n-1)(n-2).......(n-k+1)}{k!\cdot n^{k}\cdot (k+3)}$  Now How can i solve after that, Help required, Thanks I have an app installed on my phone, which I've installed from the Google Play Store.  If I update the app and don't like the upgraded version, is there a way to undo the update and downgrade back to the version of the app that I had previously installed?  I tried searching a bunch on this site, and I found lots of questions about downgrading the OS/ROM/firmware, but nothing about downgrading app versions or reverting an app update. I have a Dedicated Bedrock Server running on my PC, and I can't connect from the same PC where the server is. If I add it onto the list of servers, it appears as "Locating server" and never finds it.    If I try to connect, I get "Unable to connect to world".    On my friend's PS4, it appears on the Friends tab as a LAN Game on a Dedicated Server.    And also on my sister's laptop. Configure proxy for APT? "R cannot be resolved to a variable"? Egg replacer for bread dough? I've been reading (most) questions and answers (here) about baking bread (which I love) and how to replace eggs in many recipes (which I wish I didn't need). Turns out, I too have egg allergy (not too severe) and a few of my favorite breads require one or more eggs.  So far I've seen two options, in many threads, that look promising: Ener-G egg replacer and replacing eggs with flax. However, not a single use case I've seen here is about bread. I'm looking for an option to replace 1-3 eggs in a bread dough with: 500 g of white wheat flour, 200 ml of milk, 50-80 g of butter, a variable amount of sugar (5-50 g) and 20-30 g of fresh yeast.  Examples of this dough are the , the  and a similar recipe from Galicia (northwest Spain) with even more eggs. And so many other breads, but these are my favorite ones :)  Has anyone tried flax or ener-g on a similar bread dough?