I am revising for an exam and I am facing the following statement: $\frac{1}{1-p}=\sum_{k=0}^\infty {p^k}$  Unfortunately, I was not able to figure out why this is the case. May somebody of you quickly help me? I appreciate it a lot for a detailed explanation!
Why does the following hold:  \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ? \end{equation*}  Can we generalize the above to      $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ?   Are there some values of $x$ for which the above formula is invalid?  What about if we take only a finite number of terms? Is there a simpler formula?     $\displaystyle \sum_{n=0}^{N} x^n$   Is there a name for such a sequence?    This is being repurposed in an effort to cut down on duplicates, see here: Coping with  duplicate questions.  and here: .
Using Regex to generate Strings rather than match them How was Jon Snow able to kill the White Walker in S05E08? An Open Interval and a Half-open Interval are not Homeomorphic How to know if an object has an attribute in Python Is there a way in Python to determine if an object has some attribute?  For example:  &gt;&gt;&gt; a = SomeClass() &gt;&gt;&gt; a.someProperty = value &gt;&gt;&gt; a.property Traceback (most recent call last):   File "&lt;stdin&gt;", line 1, in &lt;module&gt; AttributeError: SomeClass instance has no attribute 'property'   How can you tell if a has the attribute property before using it? When is $A- B = B- A$? Hello I am taking a Discrete Mathematics course and having some trouble with this question about sets:  Under what conditions is $A- B = B- A$  Diagram:   Diagram:   Maybe I'm understanding incorrectly, but how can A-B and B-A be equal if they contain elements that are not in each other? Can we get notification in the tab (change of title)? I'm usually running many different StackExchange sites in Firefox using tabs (using ReloadEvery to update every 5 minutes).  Right now to check for activity, I have to manually click on the tab to see if the envelope is lighted.  How about making the icon on the tab i.e. the title text change in some way, denoting that you have activity on the page?  We already add a message count for the chat tabs, so this shouldn't be very complicated.  Possible Example: Call to a member function prepare() on a non-object PHP Help I have a list of requirements for a program, but I don't know what program meets these criteria. That's about general-purpose software, for example, not a programming IDE or statistical workbench, questions about which would certainly go to respective sites.  Is this question appropriate on Stack Overflow? Where should I ask about it? On Super User or elsewhere? How to tell Max CP? Too long in Primary Fermentation? What's the maximum amount of time for primary fermentation, assuming the beer is going straight to bottles next? Will it over ferment and then fail to carbonate, if left too long in a carboy? I want to prove that every real polynomial of odd degree has at least one real root, using  the intermediate value theorem.  Let $P(x) = x^{2n+1} + a_n x^{2n} + . . . + a_0$ for each $a_i \in \mathbb{R}$ and $n \in \mathbb{N}$.  By the fundamental theorem of algebra I know that $P(x)$ has exactly $2n+1$ complex roots, so  $P(x) = (x+r_1)(x+r_2) . . . (x+r_{2n+1})$ for each $r_i \in \mathbb{C}$  I do not know how to complete this but I do know that, at some point, I probably have to show that each root with imaginary part non zero has to come in conjugate pairs, and since $2n+1$ is odd there is at least $1$ root that is imaginary part $0$ and thus real. I'm an elected diamond moderator on one site, and a user of several others. I think it would be extremely helpful for our flagging process if users had the option to enter freeform text explaining their flag on types of flags other than the "in need of moderator attention" flag.  An example of when this is relevant: Sometimes posts have had rude/abusive flags, but the rude/abusive part might be difficult to notice, or it might be a single word or sentence buried somewhere in the middle of the post. Users have to choose between "flag it as rude with no explanation" or "flag it as misc and explain what's going on". Often they'll pick the first because of course it seems like what they should do (e.g. they know the system has special handling attached to rude/abusive flags) but it means they can't explain what's going on to us, so we have to reach out directly in private to find out.  Sometimes it also helps to be able to explain why a comment is no longer needed, because past experience tells me the "no longer needed" flag on its own might be declined on a particular site -- so I pick the custom reason instead of "no longer needed" just so I can provide explanation.  I'd prefer users be able to understand they can pick the appropriate reason and be able to attach any explanation they deem necessary to further direct moderators to the issue. If not for all kinds of flags, then at least for spam or rude/abusive flags. Solve simple trig equation Can I ever go wrong if I keep thinking of derivatives as ratios? I would like my sections to start on a new page if they do not fit on the current page. In other words:  if a section fits on the current page print it, if it does not fit start printing it on the next page.  I would like this to be done automatically.  \usepackage{titlesec} seems to be the solution, but I can not get it to do the job.   Should it be done with titlesec? how ? Is there a better approach? System.Windows.Threading.Dispatcher and WinForms? Looking for a powershell script that can pull a file from a set of PC's and FTP I'm looking to write a script (preferably powershell) that will essentially copy a file from a bunch of PC's and FTP it to a server.   So the structure of the environment is that we have a file on multiple PC's (around 50 or so) that need to placed on a server. Sometimes one of the PC's may be turned off so the script would first need to ensure the PC is up and running (maybe a ping result), then it would need to go into a directory on that PC, pull a file off of it, rename the file, place into a source directory, then remove the file. Naming convention doesn't matter, but date/time stamp would be easiest. Ideally, it would be best to first move all the files to a source directory to save on FTP bandwidth, but since the files will be named the same, the files must be renamed during the move process. Move not copy because the directory needs to be empty so the file can be re-created the next day. So once moved to the source directory, now all the files need to be FTP'd to a server for processing.   After all of this, we need to know which PC's on the list did not respond so we can manually retrieve the file so the script should output a file (txt is fine) that will show which PC's were offline.   Everything is one domain and script will be run from an server with admin creds.   Thank you! During the beta of Windows 10 it was hard know what version you were running unless it had it posted to the desktop. Once that wasn't there - how do you tell what version / build you are running?   This will become more of an issue as Microsoft starts releasing more builds with the new update mechanism. I am studying elementary Algebraic Topology recently. I have seen that a topological space is identified with a group. We are telling the group as Fundamental Group. So every topological space $X$ and for any $x \epsilon X$ there is a fundamental group $\pi_1 (X,x)$. I have a question about the converse. If we consider any arbitrary group $G$, then does there exist a topological space $X$ and an element $x\epsilon X$ such that $\pi_1 (X,x)=G$? Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$