I'm currently working on an assignment where I have to find the average case runtime of an algorithm. I've found the formula for it, which is $\sum_{i = 1}^{n - 1}\frac{1}{2}^i \cdot 2i$ which I've simplified to $f(n) = 2 \cdot \sum_{i = 1}^{n - 1}i \cdot 2^{-i}$. I'm now trying to find the closed form of the sum in the formula but I'm having a lot of trouble with it. What I do know, is that $\lim_{n \to \infty}f(n) = 4$ but I'm not sure what to do with this information. Thanks in advance for any help!
I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived.  For example, when r = 2, the formula is given by: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ according to   Thanks!
Vertical space in lists Excel function that evaluates a string as if it were a formula? Suppose I have a text string like "11+5" or even "=11+5" stored in a cell. Is there a function in Excel that will allow me to actually evaluate that string as if it were a formula?  This would be helpful for another project where I would like to be able to write 'dynamic' formulas in Excel. No association bonus for Area51? My Area51 account crossed 200 rep today, but still other accounts don't reflect the change in reputation. When will the changes be applied? or is there no association bonus for Area51 account? So, I'm using a script I've made to convert videos to the webm format. A certain program calls the script, sending %f which is the full, absolute file name of the video, like this:  converter.sh %f   where %f has two possible extensions, .avi or .mpg.  # /bin/bash ffmpeg -i $1 `dirname $1``basename $1 avi`webm &amp;&amp; rm $1   It currently works perfectly when $1 contains a .avi file, because basename removes the .avi extension. When $1 ends with .mpg instead, the result is .mpgwebm.  How can I modify that script to be able to receive those two possible different formats?    Resuming: If $1 is /somedir/video.avi, the script should do:  ffmpeg -i /somedir/video.avi /somedir/video.webm   And if $1 is /somedir/video.mpg, the script should do:  ffmpeg -i /somedir/video.mpg /somedir/video.webm   I know this might be fool for some people, but I'm kind of new with the bash. This is a reformulation of a deleted question:  If  $a_1 &gt; 0$ and $a_2 &gt; 0$ and $2a_{n+2} \le a_{n+1}+a_n$,  show that  $\lim \sup a_n \le \frac23 a_2 + \frac13 a_1$.  My proof involves showing that $a_{n+2} \le u_{n} a_{n+1} + (1-u_n)a_n $ where $u_n \to \frac23$, and I wondered if there is a simpler proof.  I will post my proof in a couple of days if no one does. I installed the WSL Bash on Windows 10 and set my user to e.g. 'abc' but now I want to change it to something else.  How can I change the user name of the user I'm logged in as?  When I open bash it automatically logs me in with my current user. What are attention mechanisms exactly? Attention mechanisms have been used in various Deep Learning papers in the last few years. Ilya Sutskever, head of research at Open AI, has enthusiastically praised them:    Eugenio Culurciello at Purdue University has claimed that RNNs and LSTMs should be abandoned in favor of purely attention-based neural networks:    This seems an exaggeration, but it's undeniable that purely attention-based models have done quite well in sequence modeling tasks: we all know about the aptly named paper from Google,   However, what exactly are attention-based models? I've yet to find a clear explanation of such models. Suppose I want to forecast the new values of a multivariate time series, given its historical values. It's quite clear how to do that with an RNN having LSTM cells. How would I do the same with an attention-based model? can I change value of $_SERVER['DOCUMENT_ROOT']? With the listings package, I would like to color the integers out of strings and comments and out of keywords too. The question for coloring numbers     . Some answer handle the numbers occurring within comments, strings or barewords, but no solution given work when a keyword contain a number. Which is annoying because C and C++ have a lot of types with a number in them, like uint32_t.  Here is a small example.  \documentclass{standalone} \usepackage{xcolor} \usepackage{listings}  \definecolor{darkgreen}{rgb}{0.18,0.54,0.34} \definecolor{maroon}{rgb}{0.64,0.16,0.16} \definecolor{darkpink}{rgb}{0.75,0.25,0.5}  \lstdefinestyle{myc++}{     language=[ISO]C++,     keywordstyle=\color{darkgreen}\bfseries,     commentstyle=\color{blue}\textit,     stringstyle=\color{darkpink}\ttfamily, %     literate=*     {0}{{{\textcolor{red}0}}}1     {1}{{{\textcolor{red}1}}}1     {2}{{{\textcolor{red}2}}}1     {3}{{{\textcolor{red}3}}}1     {4}{{{\textcolor{red}4}}}1     {5}{{{\textcolor{red}5}}}1     {6}{{{\textcolor{red}6}}}1     {7}{{{\textcolor{red}7}}}1     {8}{{{\textcolor{red}8}}}1     {9}{{{\textcolor{red}9}}}1,     escapeinside={|}{|}, %     morekeywords={int32_t} }  \begin{document}  \begin{lstlisting}[style=myc++] // In comment: 42 int32_t in_keyword = 80085; char *in_identifier|2| = "And in string 1337\n"; \end{lstlisting}  \end{document}   And the result:    The star for literate avoid to coloring the digits inside comments or strings. The escapeinside avoid coloring the digits inside a bareword when specifically marked. But I have yet to find how to color int32_t as a keyword. System Load Indicator only shows as a line unless hovered I installed the  GNOME extension on my Ubuntu 18.04, but for some reason, instead of showing a proper graph for CPU usage, it only shows a very thin line that seems to be a small portion of the actual graph. If I hover over it, the proper graph shows. How to make it show the whole graph all the time? Sci-fi series: humans are immune to telepathy, become soldiers for an alien group fighting against invertebrate telepaths The series involves two groups of aliens who have been fighting an interstellar war for many years before the first book.  The war is started by a race of jellyfish-like telepaths who have used their powers to control several species which they use as soldiers.  The jellyfish aliens have declared a holy war against all sapient vertebrates.  The only species on their side that aren't being actively controlled through their telepathy are invertebrate species who have joined the holy war.  In this series, humans are immune to telepathy--what's more, a completely unconscious reflex tends to destroy the mind of anything even attempting to affect their minds.  One of the central themes of the series is how well suited humanity is to warfare, and all the sacrifices forced on humanity as they become nothing but soldiers fighting against the jellyfish, losing almost all culture not in some way related to being soldiers, as the other races in their alliance take over the other roles.  Other species I recall include a bird race that is gifted at linguistics but can be sent into shock by even seeing a low quality recording of a human walking around a room.  They just cannot handle the hide or flee reflexes triggered by seeing such an obvious predator.  Another race is a crab like group that is so antisocial they all keep suicide buttons at hand when working with other races, and only work to help the human alliance's side because the jellyfish won't leave them alone.  There is a race that bears a faint resemblance to large rats who were previously the best soldiers on their side, and another renowned for their intellect. Statistical Entropy and Information theory When editing really long code blocks (which should definitely be refactored anyway, but that's beyond the scope of this question), I often long for the ability to collapse statement blocks like one can collapse function blocks.  That is to say, it would be great if the minus icon appeared on the code outline for everything enclosed in braces.  It seems to appear for functions, classes, regions, namespaces, usings, but not for conditional or iterative blocks.  It would be fantastic if I could collapse things like ifs, switches, foreaches, that kind of thing!  Googling into that a bit, I discovered that apparently C++ outlining in VS allows this but C# outlining in VS does not.  I don't really get why.  Even notepad++ will so these collapses if I select the C# formatting, so I don't get why Visual Studio doesn't.  Does anyone know of a VS2008 add-in that will enable this behavior?  Or some sort of hidden setting for it?  Edited to add: inserting regions is of course an option and it did already occur to me, but quite frankly, I shouldn't have to wrap things in a region that are already wrapped in braces... if I was going to edit the existing code, I would just refactor it to have better separation of concern anyway. ("wrapping" with new methods instead of regions ;) Is there a way to set transparency of windows in GNOME? Sometimes ago I used KDE 3.5 I think, and really liked the transparency option for windows, which gave all my text editor a really killer look and feel, really nice/ethereal environment for coding.  I haven't managed to find such a possibility with GNOME.  Any idea if it is possible and how? What is the mysterious stranger doing in my vault? Given that the majority of biologists do not currently consider viruses to be alive, a virus can never die.  It can, however, get destroyed by long exposures to soapy water, alcohol, and apparently certain frequencies of UV light.  Why is an individual SARS-CoV2 virion not infectious forever?  Or is it? Was the significance of Arya's Braavosi coin ever explained? Jaqen H'ghar gives Arya the iron coin, she can apparently give it to any Braavosi and say Valar Morghulis and that gets her where she needs to go, but why?    Where do these coins come from and what do they indicate to a Braavosi? I'm using 17.04 ( Clean Install ), and as far as I remember, every try to install any sofware- it fails with that error code.  I tried sudo apt-get update &amp;&amp; apt-get upgrade and problem persists.  any idea why?   edit: Using software Updater gave this result :   edit : output of update  guy@LG-LAPTOP:~$ sudo apt-get update &amp;&amp; apt-get upgrade [sudo] password for guy:  Ign:1 http://il.archive.ubuntu.com/ubuntu zesty InRelease Ign:2 http://il.archive.ubuntu.com/ubuntu zesty-updates InRelease               Ign:3 http://il.archive.ubuntu.com/ubuntu zesty-backports InRelease             Err:4 http://il.archive.ubuntu.com/ubuntu zesty Release                           503  Service Unavailable Get:5 http://security.ubuntu.com/ubuntu zesty-security InRelease [89.2 kB]      Err:6 http://il.archive.ubuntu.com/ubuntu zesty-updates Release                   503  Service Unavailable Get:7 http://archive.canonical.com/ubuntu zesty InRelease [10.2 kB]             Err:8 http://il.archive.ubuntu.com/ubuntu zesty-backports Release                 503  Service Unavailable Get:9 http://archive.canonical.com/ubuntu zesty/partner Sources [1,416 B]       Get:10 http://archive.canonical.com/ubuntu zesty/partner amd64 Packages [1,616 B] Get:11 http://archive.canonical.com/ubuntu zesty/partner i386 Packages [1,612 B] Get:12 http://archive.canonical.com/ubuntu zesty/partner Translation-en [708 B] Get:13 http://security.ubuntu.com/ubuntu zesty-security/main amd64 DEP-11 Metadata [11.7 kB] Get:14 http://security.ubuntu.com/ubuntu zesty-security/main DEP-11 64x64 Icons [9,219 B] Get:15 http://security.ubuntu.com/ubuntu zesty-security/universe amd64 DEP-11 Metadata [14.4 kB] Get:16 http://security.ubuntu.com/ubuntu zesty-security/universe DEP-11 64x64 Icons [31.0 kB] Reading package lists... Done                                          E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty-updates Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty-backports Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. guy@LG-LAPTOP:~$    output of apt get  guy@LG-LAPTOP:~$ sudo apt-get install putty Reading package lists... Done Building dependency tree        Reading state information... Done The following additional packages will be installed:   putty-tools Suggested packages:   putty-doc The following NEW packages will be installed:   putty putty-tools 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 661 kB of archives. After this operation, 2,713 kB of additional disk space will be used. Do you want to continue? [Y/n]  Err:1 http://il.archive.ubuntu.com/ubuntu zesty/universe amd64 putty-tools amd64 0.67-3   503  Service Unavailable Err:2 http://il.archive.ubuntu.com/ubuntu zesty/universe amd64 putty amd64 0.67-3   503  Service Unavailable E: Failed to fetch http://il.archive.ubuntu.com/ubuntu/pool/universe/p/putty/putty-tools_0.67-3_amd64.deb  503  Service Unavailable E: Failed to fetch http://il.archive.ubuntu.com/ubuntu/pool/universe/p/putty/putty_0.67-3_amd64.deb  503  Service Unavailable E: Unable to fetch some archives, maybe run apt-get update or try with --fix-missing? guy@LG-LAPTOP:~$ Comma after introductory words, phrases, clauses: unacceptable, obligatory or optional? Permutations with repeating digits My question is this : how many distinct two digit numbers can be produced from numbers $4, 3, 3, 1$? When applying the formula $$\frac{4!}{(4-2)!2!}$$ you come up with $6$, yet when doing the problem manually, I come up with $7$ numbers, namely $41, 43, 14, 13, 33, 31, 34$. Can anybody explain this discrepancy?  Thanks.