I would like to compute the value of this series:  \begin{equation*} \sum_{n = 0}^{+ \infty} n . e^{- \alpha n} \end{equation*}  Where $\alpha$ is a constant.
I'm trying to compute the infinite series $\sum_{n=0}^{\infty} ne^{-n}$.  I know the answer is $e/(e-1)^2$, but I don't understand how to find this result.  Thanks for the help!
Where does the idiom "beating around the bush" come from? Where does the idiom "beating around the bush" come from? What is DOCTYPE? What is DOCTYPE and why do I want to use it? What are the different DOCTYPEs I can use? What is the difference between standards and quirks mode, and what are some quirks I may run into with differently set DOCTYPEs?   Lastly, what is the proper DOCTYPE that I should be using? I commonly use the word 'guys' to refer to a group of males colloquially. It's colloquial but not rude, off putting, condescending, patronizing (though I wouldn't use it with a group of men at a board meeting (hypothetically of course), unless I knew them). So, some that I would not consider as replacements would be    buddies: too old-fashioned dudes: too informal boys: too patronizing men, gentlemen: too formal or false respect   I realized that I had also been using it for any group of people, males and females, even just females. It worked for me and I didn't think of the gender implications, that women might not care for it. No one ever complained, directly or indirectly (rule of life: sometimes people don't complain about things they should).   But it occurred to me (maybe by reading something that sparked realization) that the intended hearers might not all care for it.  What might be a female gendered or non-gendered version of 'guys'?  Some that I've considered don't feel right about (though these may be reasonable answers) are:   dolls: too old-fashioned babes: too informal  ladies: too formal or I've heard from women, too creepy women: too factual girls: too patronizing (as much as 'boys') everyone, you all: too bland y'all: perfectly non-sexist but too regional (might work, but not everywhere)   Hopefully I haven't eliminated all the possibilities -- any suggestions? What’s the rule for using “who” and “whom” correctly? Gluing Lemma for Closed Sets: Infinite Cover Counter-Example Appositives with possessiveness? I would like to say something like     The boy, Adam's, favorite toy was a bike.   What is the proper way to say this? Can the coefficient of static friction be less than that of kinetic friction? Describe in English a Turing machine that semidecides the following language - Is this solution correct? The language being L, which is described as follows:  $L$ = { $&lt;M&gt;$ | $M$ accepts the binary encodings of at least 4 odd numbers } .  My solution:  M generates the binary encodings in $\Sigma_M*$ in ascending order and uses dovetailing to interleave the computation of M on those binary encodings. As soon as four computations accept, M halts and accepts.  Is this correct? What corrections would have to be made if not? I am not sure if I am providing enough details in my solution which may cause it to be incorrect. How to choose the right bean scope? Postgres: Reclaiming Space old Designjet 430 with Windows 10 SVM for unbalanced data I want to attempt to use Support Vector Machines (SVMs) on my dataset. Before I attempt the problem though, I was warned that SVMs dont perform well on extremely unbalanced data. In my case, I can have as much as 95-98% 0's and 2-5% 1's.   I tried to find resources which talked about using SVMs on sparse/unbalanced data, but all I could find was 'sparseSVMs' (which use a small amount of support vectors).  I was hoping someone could briefly explain:   How well SVM would be expected to do with such a dataset   Which, if any, modifications must be done to the SVM algorithm What resources/papers discuss this How can I switch back to Nvidia card from Intel with Nvidia-prime So I updated to Ubuntu 14.04 and found that the primusrun command no longer works, so ditched bumblebee in favour of nvidia-prime. This lets you switch video card from nvidia-settings rather than running through the command line (I prefer the old method, but want the primus performance).  The problem is that after switching to Intel from nvidia-settings, I can no longer open it up again to switch back to the nvidia card. When I open nvidia-settings I get a segmentation fault (according to the terminal).  Is there any way to open nvidia-settings switch back? I would like the ability to be able to switch between the two since having the nvidia card on all the time kills my battery and heats up the laptop a lot. What is the white smoke-like fog called? As I navigate through physics stackexchange, I noticed .  Electrons radiate light when revolving around nucleus? Why is it so obvious?  Note that I do not know anything about quantum mechanics - and I just want classical stories. A locally constant sheaf on a locally connected space is a covering space; Proof? As part of my hobby i'm learning about sheaves from Mac Lane and Moerdijk. I have a problem with Ch 2 Q 5, to the extent that i don't believe the claim to be proven is actually true, currently. Here is the question repeated:     A sheaf on a locally connected space $X$ is locally constant if each point $x \in X$ has a basis of open neighborhoods  $\mathcal{N}_x$ such that whenever $U,V \in \mathcal{N}_x$ with $U \subset V$, the restriction $\mathcal{F}V \rightarrow \mathcal{F}U$ is a bijection. Prove that $\mathcal{F}$ is locally constant iff the associated etale space over $X$ is a covering.   I don't have a problem (i think) showing the "covering" $\rightarrow$ "locally constant" direction of the implication. My problem is in the reverse direction. In fact, in the process of trying to solve this problem i think i have come up with a very simple counterexample.  Here's my attempt at a counterexample:  Let $X = \{a,b,c\}$ be  finite space with open subsets $\{\{\},\{a\},\{a,b\},\{a,b,c\}\}$. I claim this is hyper-connected and locally connected.  Let $\mathcal{F}$ be the sheaf with $\mathcal{F}(\{a,b,c\}) = \{q\}$, $\mathcal{F}(\{a,b\}) = \{r,s\}$, and $\mathcal{F}(\{a\}) = \{t\}$, where $r = q|_{\{a,b\}}$, and $t = q|_{\{a\}} = r|_{\{a\}} = s|_{\{a\}}$. I claim this defines a locally constant sheaf. Indeed we can have $\mathcal{N}_{a} = \{\{a\}\}$, $\mathcal{N}_{b} = \{\{a,b\}\}$, and $\mathcal{N}_c = \{\{a,b,c\}\}$.  Now we can calculate the corresponding etale space $p : E \rightarrow X$ as follows: $E = \{\dot{q}(a), \dot{q}(b), \dot{q}(c), \dot{s}(b)\}$, and has these four distinct elements since $\dot{q}(a) = \dot{r}(a) = \dot{s}(a) = \dot{t}(a)$, and $\dot{q}(b) = \dot{r}(b)$. The action of $p$ follows by construction.  However, $p$ does not appear to me to be a covering space. In particular, the only neighborhoods of $b \in X$ are $\{a,b\}$and $\{a,b,c\}$ and we have $p^{-1}(\{a,b\}) = \{\dot{q}(a),\dot{q}(b),\dot{s}(b)\}$ and this cannot be partitioned into homeomorphic images of $\{a,b\}$ because $2 \nmid 3$. Similarly, $p^{-1}(\{a,b,c\}) = \{\dot{q}(a), \dot{q}(b), \dot{q}(c), \dot{s}(b)\}$ and this cannot be partitioned into homeomorphic images of $\{a,b,c\}$ since $3 \nmid 4$.  So i think i have a locally constant sheaf on a locally connected space whose corresponding etale space is not a covering. What have i done wrong? I have seen the claim mentioned elsewhere so i'm convinced i'm wrong somewhere. Perhaps someone knows where a correct proof is published otherwise? Chapter number starts at 2 not 1 How can I start my chapter from "Chapter 2" instead of "Chapter 1".  I am preparing LaTeX document for each chapter. when I put \chapter{Introduction}, it produces "Chapter 1" Introduction. However I want it to be "chapter 2" Introduction instead. Wouldn't it be a good idea to be able to vote for questions to be moved to any Stack Exchange site? For example, I just voted  as off topic, but that was all I could do. I wish I could've voted it be moved to .  It's OK to have the predefined list of sites that is now when voting to move, but it would be nice to have a more link, where we could pick any Stack Exchange site.  Here's a quick mockup of how I see this implemented:    When the more link is clicked, a list of all the Stack Exchange sites should appear. How to show that the set of orthogonal n x n matrices forms a group under multiplication I am studying orthogonal matrices and I am not sure if to show if a set of orthogonal $n \times n$ matrices forms a group under multiplication. We must check each of the group axioms. I found that the axioms are:  Closure Associativity Existence of identity matrix Existence of the inverse matrix.  I edited my question, since I was able to find more information about this topic. This group is called $O(n)$. To check the four axioms I did: Let $A \text{ and } B \in O(n)$, denoted as orthogonal matrices and assume that $C=AB$, then, Closure :  To prove that $C \in  O(n)$ we must prove that $C$ is a real $n \times n$ orthogonal matrix with uni-modular determinant. Since A and B are real $n \times n$ matrices, $C$ is also a real $n \times n$ matrix so, $C^TC=(AB)^T AB=B^T A^T AB = B^TB=I$  Associativity :  Matrix multiplication is associative, so the law holds for $O(n)$ group elements.  I am not sure if this is enough to prove associativity. Identity element :  The $n \times n$ identity matrix $I_{n \times n}$ represents the identity element.  In this case I am not sure if this is enough to prove the identity element. Inverse element :  Let $A^{-1}$ be the inverse of $A$, then we need to prove that  $A^{-1} \in  O(n)$ since $(A^{-1})^T=(A^T)^{-1}$. We have that: $(A^{-1})^T A^{-1}=(A^T)^{-1} A ^{-1}=(AA^T)^{-1}=I^{-1}=I$  Can anyone check if what I did is correct? I also would like to know if I can prove the associativity and the identity element in a better way. thanks How do I limit the running time of a BASH script I have a long running BASH script that I am running under CYGWIN on Windows.  I would like to limit the script to run for 30 seconds, and automatically terminate if it exceeds this limit. Ideally, I'd like to be able to do this to any command.  For example:  sh-3.2$ limittime -t 30 'myscript.sh'   or   sh-3.2$ limittime -t 30 'grep func *.c'   Under cygwin the ulimit command doesn't seem to work.   I am open to any ideas.