Summing $\sum_{n=0}^{\infty} n x^n$
Sum of a power series $n x^n$
Via command line, I have a log file I'd like to keep track of.  What I want is to have, basically, a tail that refreshes when the log is updated making the text scroll upwards as new lines are appended to the log file.  Is there anything out there that does that without having to write some code? alias multi commands as one alias I made an alias for youtube-dl  with some options  like that  yd="youtube-dl -cit -f 18"   so I can download it like   yd URL   I want to modify it to show only the two lines from the output using   grep -e "Downloading video #" -e ETA  I can make it in two aliases using the |   but is there a way to make it one alias Why is $\mathbb{E}[X] = 1 + \sum^\infty_{k=1}\mathbb{P}(X > k)$ true? I'm working through a problem regarding expected values in Markov chains, and at some point it says:     Recall from probability that if $X$ is a positive integer valued random variable, then $\mathbb{E}[X] = 1 + \sum^\infty_{k=1}\mathbb{P}(X &gt; k)$.   I know that by definition $\mathbb{E}[X] = \sum_a a\mathbb{P}(X = a)$, but I can't see how the above equality follows from this, nor am I sure if this is how to approach the problem. What causes the expansion of space? Neural networks output probability estimates? Suppose my training data contains ~100 variables, and each example is tagged as "success" or "failure".  I understand how a neural network can be used to try and predict success vs failure based on the variables.  However I am interested in the neural network outputting the posterior probability rather than success or failure. In fact, I evaluate the efficacy of the NN based on how accurate the probabilities are (eg. AUROC over the entire dataset) rather than % of cases with correct prediction.  Are NN's the right tool for the job here and if so how do you structure the NN to output this?  (NOTE: I'm a ML newbie!) I am using Ubuntu 11.10 and my screensaver keeps coming on while I'm watching a movie.  I don't remember this happening in 11.04 and can't remember how to stop it? I am building a drupal site using Views, search API module and user-points module.  I created the "database server", and indexed many fields of nodes.  I created a search page using views and the search API, that present content (image, title etc.) In addition, I want to add to every content in this search view the name of the author that wrote it, his author picture.  I add the Indexed Content: Author relationship. Unfortunately, I can't find a way to add the author picture and the current total point to my view.  Can somebody advice how to do that? Show rigorously that the sum of integrals of $f$ and of its inverse is $bf(b)-af(a)$ Suppose $f$ is a continuous, strictly increasing function defined on a closed interval $[a,b]$ such that $f^{-1}$ is the inverse function of $f$. Prove that,   $$\int_{a}^bf(x)dx+\int_{f(a)}^{f(b)}f^{-1}(x)dx=bf(b)-af(a)$$   A high school student or a Calculus first year student will simply, possibly, apply change of variable technique, then integration by parts and he/she will arrive at the answer without giving much thought into the process. A smarter student would probably compare the integrals with areas and conclude that the equality is immediate.  However, I am an undergrad student of Analysis and I would want to solve the problem "carefully". That is, I wouldn't want to forget my definitions, and the conditions of each technique. For example, while applying change of variables technique, I cannot apply it blindly; I must be prudent enough to realize that the criterion to apply it includes continuous differentiability of a function. Simply with $f$ continuous, I cannot apply change of variables technique.  Is there any method to solve this problem rigorously? One may apply the techniques of integration (by parts, change of variables, etc.) only after proper justification.  The reason I am not adding any work of mine is simply that I could not proceed even one line since I am not given $f$ is differentiable. However, this seems to hold for non-differentiable functions also.  I would really want some help. Pictorial proofs and/or area arguments are invalid. Badge progress reports Is there a way to view your own progress towards certain badges? For example:   Tumbleweed - What value constitutes low views for a week? Fanatic  – Is there a way to view how many days in a row you currently have? Beta – Is it even possible to get this badge anymore? How to install "vanilla" TeXLive on Debian or Ubuntu? Debian and Ubuntu have very good texlive packaging. But I'd like to use tlmgr to have a more fine-grained control about upgrades and which packages I want to be installed.   So how do I install "vanilla" TeXLive on Debian/Ubuntu?   Also how do I make dpkg aware that I have TeXLive installed? `du -sh` reports different used size than `df -h` Possible Duplicates:             df -h says that I have 494G used on /var:  Filesystem            Size  Used Avail Use% Mounted on /dev/sda3             564G  494G   41G  93% /var   But du -sh /var says it's 350G  What makes them different? I mistakenly typed cd // instead of cd /. To my surprise, current directory showed up as //.   What is that directory? Why does it exist?  apple@snipped $ pwd /home/apple apple@snipped $ cd / apple@snipped $ pwd / apple@snipped $ cd // apple@snipped $ pwd // apple@snipped $ cd /// apple@snipped $ pwd / Prove This Inequality ${\pi \over 2} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}} \le {\pi \over 2} + 1$ $${\pi \over 2} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}} \le {\pi \over 2} + 1$$  I see I should use Riemann sum, and that $$\int_0^{\infty} {dx \over {1+x^2}} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}}$$  But how do I explain this exactly, and how to get the other half of the inequality (with ${\pi \over 2} + 1$)? bi-directional sync with rsync I am running Ubuntu 14.04 and Notepad++ 6.6.4 (through WINE)  Under the 'Run' tab, I clicked on "Launch in Firefox" , but nothing happened   I uninstalled Notepad++ and WINE, then shutdown. I re-installed Notepad++ and WINE and still have the same problem.   Does anyone know what the problem is? Find a bijection to prove a fact about integer partitions Prove that for fixed positive integers $k$ and $n$, the number of partitions of $n$ is equal to the number of partitions of $2n + k$ into $n + k$ parts.  I tried this with different values of n but can't seem to get a bijection between the two partitions. What is the best way to include R code in the Latex paragraph I am a newbie in Latex. I have two simple questions regarding including R code in Latex.  Q1: For a block of R code, I use 'listings' package, as follows.  \documentclass[12pt]{report} \usepackage{listings} \begin{document} \begin{lstlisting}[language=R] &gt; dpareto(4, 2, 3) [1] 0.09375 &gt; dpareto(4, -2, 3) [1] NaN &gt; dpareto(seq(-1, 4, 1), 2, 3) [1] 0.0000000 0.0000000 0.0000000 0.0000000 0.2962963 0.0937500 \end{lstlisting} \end{document}   I feel it's not professional. What is the best way to present the block of R code?  Q2: For code pieces in the paragraph, for example I wish to include a R function, or variable, or package name in my Latex context, what is the best way to do that? Can you trip as part of a Tome of Battle Maneuver? If a ToB Maneuver such as Daunting Strike calls for a melee Attack as part of the maneuver, can this melee attack be resolved as a trip attack/include a trip attempt? I have asked this question before  The first guy solved it in a way, but  i would like to know is there any efficient way to do so, because i did the same for N = 13 and N = 15  which took long time.  This problem is from a pen and paper test, ZIO-2018 Can i solve it with an efficient algorithm, well i am pretty beginner in algos, so if there is one kindly descibe it how can i do so?  Thank you How can I get Stack Exchange to send me to a conference? I read , which states,  Depending on the circumstances and location, we can also sponsor community leaders to attend an event on behalf of their site. We will subsidize your costs to attend, within reason, and provide you with a bunch of swag to use as an ice-breaker when introducing yourself.  This sounds great. But the blog is sparse on the details. What do I actually have to do to get in on this?