If the $n^{th}$ partial sum of a series $\sum a_n$ is $S_n = \frac{n−1}{n+1}$ then find $a_n$ and find the sum of this series. Fully justify your answer.  Using the definition.  $a_n = S_{n} - S_{n-1} $  $$a_n = \frac{n-1}{n+1} - \frac{n-2}{n}$$  $$= \frac{n(n-1) - (n-2)(n+1)}{n(n+1)}$$  $$= \frac{n^2-n - (n^2-n-2)}{n(n+1)}$$  $$= \frac{2}{n(n+1)}$$  Hmm, I can use PFD on this to make it.   $$= \frac{2}{n} - \frac{2}{n+1}$$  Is this right? What about n = 0   The sum is 1 right? cus lim at $\infty$ ?
If the $n^{th}$ partial sum of a series $\sum a_n$ is $S_n = \dfrac{n-1}{n+1}$, find $a_n$ and the sum.  By definition,  the sum of the series is the $\lim n\to\infty$ of it's $n^{th}$ partial sum.  $$\text{ Sum = } \lim_{n\to\infty} = \dfrac{n-1}{n+1} = 1$$  I am asked to find $a_n$. How do I do this? What is the procedure?
I was writing some Unit tests last week for a piece of code that generated some SQL statements.  I was trying to figure out a regex to match SELECT, INSERT and UPDATE syntax so I could verify that my methods were generating valid SQL, and after 3-4 hours of searching and messing around with various regex editors I gave up.  I managed to get partial matches but because a section in quotes can contain any characters it quickly expands to match the whole statement.  Any help would be appreciated, I'm not very good with regular expressions but I'd like to learn more about them.  By the way it's C# RegEx that I'm after.  Clarification  I don't want to need access to a database as this is part of a Unit test and I don't wan't to have to maintain a database to test my code. which may live longer than the project. How can I pay tolls on Japanese expressways? I'm renting a car soon in Japan and am planning to travel on the expressways. What's the best way to pay tolls? How can I copy the path to a folder or a file? The Info Dialog doesn't allow me to copy the information that stands there ...    Is there an easy way to do this? If $A$ commutes with $B$ and $B$ commutes with $C$, then $A$ commutes with $C$ Let $A,B,C \in \ M(3,\mathbb{R})$ be such that $A$ commutes with $B $and $B$ commutes with $C$. $B$ is not scalar matrix. Then, $A$ commutes with $C$.   Is this true or false? I think it is not true, but cannot think of a counterexample. Any suggestions? I started using Sublime Text 2 a week or so ago and I loved it so much I've spent $60 on it and now want to use it for everything.  Gedit currently has ownership of most of the text file associations. I'd essentially like to replace any association to gedit with sublime-text-2.  Any tips available on bulk association changing? Desktop-neutral methods are preferred. Java Command line arguments Logical Volume, Physical Volume and Volume Groups Please can someone explain to me the meaning of Logical Volume, Physical Volume and Volume Groups? Of what importance and advantage are they in linux? I have equations inside an environment which has a larger left-margin than the main text (analogous to an enumerate/itemize environment, implemented with the changepage package's adjustwidth environment). Just as equations in enumerate/itemize behave, the equations are centered with respect to the entire text, rather than just the width of the container environment. I want the latter behavior.  I'm trying to accomplish this by wrapping all display-math in minipages. Namely, I'm modifying the mathdisplay environment in amsmath.sty as follows:  \makeatletter \let\oldmd\mathdisplay \def\mathdisplay#1{%   \newline   \begin{minipage}{\textwidth-\@totalleftmargin}%     \oldmd{#1}% } \let\oldendmd\endmathdisplay \def\endmathdisplay#1{%   \oldendmd{#1}%   \end{minipage} } \makeatother   This is almost working, but there are some issues with spacing which I don't understand how to fix. Here's how the the normal \[ ... \] and numbered \begin{equation} ... \end{equation} look with the above modification:      My main confusion is why the numbered version is behaving so differently from the unnumbered. But all in all I want to fix this so that the spacing above and below is even as it usually is. Any help is appreciated, thanks.  Edit: Here's a compilable example:  \documentclass{amsart}  \usepackage{lipsum} \usepackage{calc} \usepackage{changepage}  \makeatletter \let\oldmd\mathdisplay \def\mathdisplay#1{%   \newline   \begin{minipage}{\textwidth-\@totalleftmargin}%     \oldmd{#1}% } \let\oldendmd\endmathdisplay \def\endmathdisplay#1{%   \oldendmd{#1}%   \end{minipage} } \makeatother  \setlength{\parindent}{0pt}  \begin{document}  \lipsum[1]  \begin{adjustwidth}{20pt}{} \lipsum[2] \[ x \mapsto \hom(-,x) \]  \lipsum[3] \begin{equation}   y \mapsto \hom(y,-) \end{equation} \lipsum[4] \end{adjustwidth}  \lipsum[5]  \end{document} Why can't I travel onwards if my bag wasn't going to make the flight? I was flying KLM from the UK to the USA with a short layover in Amsterdam.  Because of delays in the UK, I almost missed my flight.  The flight attendant told me my bag wasn't going to make it, but if I walk fast, my gate is just two over.  When I got to the gate, they had already closed the doors, but the counter was opened - I mentioned that my bag was not going to make it, but that's okay.  She said, "hold on," got onto her walkie-talkie, and said,  "Yep, that's what I thought: you can't fly if your bag's not going."  I originally thought that it might be a security concern, but that doesn't really make any sense.  Why can't I travel if my luggage doesn't make the connection?  EDIT: I should clarify that I have no beef with KLM - they treated my overnight in Amsterdam professionally and helpfully; I'm just curious what the reason would be. Are \( and \) preferable to dollar signs for math mode? Portuguese tab names on profile page A few minutes ago, when I viewed profile pages on my phone, the tabs were all in Portuguese:   On the mobile interface:   On the full site     But it didn't happen on my desktop, and now it's back to normal.  Other users report that the site is normal, but  says it's still happening, so I've decided to post this bug report even though it seems to be working for me at the moment.   Perhaps it's some sort of intermittent problem.  I'm posting this on Meta.SE, since it seemed to happen on both Japanese.SE and ELL.SE, so it doesn't seem to be specific to a particular site. Whenever I access Stack Exchange the tab appears busy What character / attack / stage combination can get me a combo of 100? For the first two challenge panels, I just used my golden hammers on the "get a combo of x," but now I'm seeing on challenge panel 3, the game expects you to get a combo of 100! I can barely get past a combo of 9 without the counter resetting. What character / stage combination can get you a combo of 100? Are there multiple, so I can more easily get the "get a total combo of 400 across all characters" challenge?  Additionally, what exactly constitutes a combo? Sometimes my combos are cut short and sometimes they're longer, so I must not understand how they're tracked. Inequality from Challenge and Thrill of Pre-college Mathematics How to render from Eevee in Blender 2.8? I have the Blender 2.8 version from the 21st Sep 2018 I was looking forward to render a test sequence but cannot find a render button anywhere? Does anyone know where to find it? What happened to the "very low quality" flag reason? Possible Duplicate:          I see a  to receive a "very low quality" flag by me.  It's just a link pointing elsewhere basically saying, "try this." I generally do this for answers I see that are like this and have been successful every time so far.  I save "not a real answer" flag for other cases such as someone asking another question in an answer or "thanks" answers.  I want to flag it as "VLQ" but I don't see that as an option*.  In fact, I don't see it as an option anymore on any site throughout the Stack Exchange network.  * This was not possible before when I initially posted this. Michael's observation on the needed downvote seems to be the reason it didn't show up.  Did it just get phased out as being unnecessary?  Should I be using "NARA" as the flag reason for this case now?  I'm not sure if the "NARA" explanation really covers this case well: (emphasis mine)     This was posted as an answer, but it does not answer the question. It should possibly be an edit, a comment, another question, or deleted altogether.   These sorts of answers can answer the question in general.  The site linked to might very well completely answer the question.  (in this particular case, maybe not however)  It's just that the answerer really should do something to improve the answer.  It's not like it's a "This question is a duplicate of this [onsite question] post" or some other general spam.  Those cases are clearly "NARA"/spam.  There was a  regarding the necessity of this flag reason.  Answers point out that it is needed and I agree.  There hasn't been anything mentioned (that I am aware of) in terms of removing it by the team.  Was this officially removed now in favor of other flagging reasons?  Or is this just a bug and it was "accidentally" removed? Find $P(12)$ when $P(x)={1\over(x+1)}$, for $x=0,1,2,......11$ let $P(x)$ be a polynomial of degree $11$ such that  $P(x)={1\over(x+1)}$, for $x=0,1,2,......11$  then find value of $P(12)$ How does linear regression use the normal distribution? In linear regression, each predicted value is assumed to have been picked from a normal distribution of possible values. See below.  But why is each predicted value assumed to have come from a normal distribution? How does linear regression use this assumption? What if possible values are not normally distributed? Cannot authenticate without a password Is the empty set a group?