Partial sum back into series.
If the $n^{th}$ partial sum of a series $\sum a_n$ is $S_n = \frac{n-1}{n+1}$, find $a_n$ and the sum.
I am trying to define an alternative \item command that would allow me to use a different color for some of the items. My code has the following form:  \documentclass{article} \usepackage{color}  \definecolor{gray}{rgb}{0.7,0.7,0.7} \newcommand{\grayitem}[1]{{\color{gray} \item #1}}  \begin{document}  \begin{enumerate} \item First item. \grayitem Second item. \item Third item. \end{enumerate}  \end{document}   I was hoping that  \grayitem Second item.   would produce the same result as  {\color{gray} \item Second item.}   but for some reason, these are not equivalent. They produce the following results, respectively:  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;   Why are these two lines of code not equivalent? What am I doing wrong and how can I change the definition of \grayitem to produce the second result?  I have also tried removing a pair of braces, like this  \newcommand{\grayitem}[1]{\color{gray} \item #1}   but then the third item becomes gray as well.  The solution should also work for multi-paragraph items. Please consider gathering Meta feedback before the "ship" stage Derivative of function solved from NDSolve can't be plotted I have my Git repository which, at the root, has two sub directories:  /finisht /static   When this was in , /finisht was checked out in one place, while /static was checked out elsewhere, like so:  svn co svn+ssh://admin@domain.com/home/admin/repos/finisht/static static   Is there a way to do this with Git? I edited my mesh changing the number of vertices after adding shapekeys, and it get totally messed up when the shapekeys are on. Is there a way to solve this problem without remake the shapekeys all again!? Powerset of set with empty sets I am working on a problem with the following set:  $$S =  \{\varnothing,\{\varnothing\}\}$$  My solution was: $P(S) = \{\varnothing, \{\varnothing\},\{ \varnothing , \{ \varnothing \}\}\}$, but the solution on the textbook shows: $P(S) = \{ \varnothing , \{ \varnothing \}, \{\{\varnothing\}\}, \{ \varnothing , \{ \varnothing \}\}$  Where does the $\{\{\varnothing\}\}$ come from?  Thanks What does (obj.length === +obj.length) compare? HD bad blocks: is there a way to repair them? How do I parse an ISO 8601-formatted date? Are there constants in JavaScript? Is there a way to use constants in JavaScript?  If not, what's the common practice for specifying variables that are used as constants? Would Gravitational Time Dilation still apply between two gravitational sources? As explained on , when an object is experiencing strong gravitational forces, time passes slower for it. If, in theory, you had two objects exerting the same amount of gravitational pressure on an object in opposite directions, effectively cancelling each other out, would they cancel out the time dilation, or would they compound, making the time dilation more prevalant? Can a script modify an environment variable of its calling shell? I am setting my JAVA_HOME in my ~/.bashrc file.  Now, I need to create a script that will change the JAVA_HOME just for the current terminal, so that I can start an app that needs java 5.  I have created this script to do this task, but after finishing, I can see that JAVA_HOME is not updated  export JAVA_HOME=/usr/lib/jvm/java-5-oracle/ export PATH=$PATH:$JAVA_HOME       Here is the result  $ ./javaHome5.sh $ echo $JAVA_HOME /usr/lib/jvm/java-6-oracle/   I think that the reason this is not being applied, is that a script is executing in it's own terminal, so when the script ends, the current terminal will not be affected.  Currently, the only way I've found around this is:   Edit my ~/.bashrc and change the JAVA_HOME var Run source ~/.bashrc to apply the changes in current terminal. Which again cannot be applied in a script, as the source command needs to be run in the current window.   Needless to say, this change applies to all new terminal windows, so I practically need to do this twice: One before starting my app, and one more time right after this, just to restore the environment vars to their default. That's not really convenient.  Do you have any ideas how can I change this var using a script? What is a plain English explanation of "Big O" notation? I'd prefer as little formal definition as possible and simple mathematics. How to install KDE along Unity? I have just installed the latest Ubuntu 14.04 and I would like to install KDE along side my Gnome/Unity that I have now. How can I do this?  Thanks! Why didn't the Ministry use owls to find Sirius? Harry, before the Triwizard Tournament, was able to send owls to Sirius when he was in hiding (not at Grimmauld Place) just by telling them "just find him, will you?", and the owl would come back a few days later with an answer. This doesn't happen with Hedwig only, because, if I recall correctly, he uses school owls as well.  If finding Sirius was as easy as telling an owl "find him", and a 13-year-old wizard could do it, how come the Ministry had no clue as to where he was? They could've sent an owl with a summons and followed it from a distance.  Two theories, neither mentioned in the book, and both a bit unlikely:   He was living as a dog and -hopefully- the Ministry would not be able to recognize him. Unlikely because the owl would have got there anyway, so they would have guessed the dog was Sirius in disguise, not necessarily as an animagus, but maybe he could perform human-animal transfiguration (remember he was brilliant at school).. Kingsley Shacklebolt was already in charge of the hunt and Dumbledore told him Sirius was innocent and all that. Then, Kingsley would have prevented the Ministry from using any method that could've actually worked. Unlikely because they could've used owls before Dumbledore's tip, when Kingsley didn't know Sirius was innocent, and also because Kingsley was quite powerful and had the support of Crouch and others at the Ministry, so if he was in charge, they would probably have succeeded.   Your thoughts? Selecting ARIMA orders based on ACF-PACF vs. auto.arima I have a valid B1/B2 visa valid to US till March 2019 on Indian passport, I changed my citizenship to Australian. Is my B1/B2 visa on Indian passport valid? Prove that every terms of a sequence defined by a recurrence relation is a perfect square. What licence is subscriber content provided under 4.0, 3.0 or them all? What spell targets count for the Graviturgy Magic wizard's Gravity Well feature? The  feature from the  (EGtW, p. 185) triggers &quot;whenever you cast a spell on a creature&quot;, and has an effect &quot;if the target is willing to move, the spell hits it with an attack, or it fails a saving throw against the spell.&quot; It's not clear to me what kinds of spells can actually trigger this feature. Based on the singular phrasing of &quot;on a creature&quot; and the later use of the word &quot;target&quot;, I would normally read this as only applying to spells that explicitly target creatures (and perhaps only those with a single target, in contrast to the phrasing of the  feature of the  wizard, for example). However, spells that target a location can use the word target in a different sense (see ). What's actually going on here? Does this feature only work on targeted spells? Or does it function for area of effect spells as well? Or is this just an ambiguous phrasing that will rely on the DM's ruling?