We know that every planet in our solar system revolve's around the sun in a particular orbit. But were to they get the energy to revolve around the sun. And why do they not drop into the sun there is only gravitational force acting which is always attractive in nature?
Why doesn't the Moon fall onto the Earth? For that matter, why doesn't anything rotating a larger body ever fall onto the larger body?
Problem regarding radius of convergence of $\sum n!x^{n^2}$. $$\sum_{n=0}^\infty n!x^{n^2}$$   Can anyone tell me what will be the radius of convergence for this power series. According to me it is $0$.   If you can please show me how to go about it if I am wrong. Story is in a medievalish setting and starts with a woman who has been falsely accused of being a witch being taken by 2 men to be sacrificed to a dragon.  She ends up befriending the (male) dragon, who is able to transform into various forms, including human, and can speak only the language of whatever form he is in. e.g. can speak English in human form, but can't in dragon form.  He needs to return to his dragon form every day at sunrise or he dies.  Contact with iron or steel causes him pain and prevents him from transforming.  Near the end of the book, he's been put in handcuffs by one of the two men from the start and the woman has to get the assistance of the other one (village blacksmith?) to remove them before sunrise, which they manage to do just in time.  I think I read it maybe 10 years ago.    I also remember it was a very short book, probably only about 100 pages. When should I use velocity versus addForce when dealing with player objects? Why 220V and 110V? Why did the industry choose these seemingly unintuitive values of 110V and 220V for the mains AC? Is there a physical reason why they aren't simply 100V and 200V? I've sectioned of this area in the sea near my base, and I need to get rid of all the water from it.  From what I've tried the only way I can see to clear it is fully cover one layer, delete the blocks, then go down to the next layer. Is that the only way I can clear the area or is there a better approach?  Ideally I'd like a survival method (if there is a better one), but any commands what could be used I'd like to hear just so I can learn more commands :) Apache and systemd I've just started using Debian 8 which has introduced systemd. I'm not interested in the politics around SysV vs systemd, but I am confused, especially around Apache (2.4).  There are now two ways to restart apache:   apache2ctl restart or apache2ctl graceful systemctl restart apache2 and it seems systemctl restart apache2.service does the same thing.   And they appear to do different things(!). I don't understand the difference.  I seem to need to do both to get a proper full restart working. e.g. after changing a config file for the php module it only noted the change after both.  Another time when apache2ctl configtest was failing and the failure was reported in systemctl status apache2 I fixed the config so that the first of these commands was happy, restarted with apache2ctl restart but the systemd status command still listed it as not running and with the old config problem.  I promise to spend ages reading about systemd (RTFM) in depth at some point, but meanwhile on a practical level relating to apache, I'd appreciate advice. I have a vector space $V$ over a field of characteristic $0$. If $M_1,\dots,M_k$ are proper subspaces of $V$, and $N$ is a subspace of $V$ such that $N\subseteq M_1\cup\cdots\cup M_k$, how can you tell $N\subseteq M_i$ for some $i$?  I was first trying to show it just in the case $N\subseteq M_1\cup M_2$, and hoping to extend it to finitely many $M_i$. If either of the $M_i$ contains the other, the claim follows, so I suppose neither $M_i$ contains the other.   In hopes of a contradiction, I suppose $N\not\subseteq M_1$ and $N\not\subseteq M_2$, Picking $x_1\in N\setminus M_1$ and $x_2\in N\setminus M_2$, I'd have $x_1+x_2\in N\subseteq M_1\cup M_2$. The only thing I could conclude was that actually $x_1,x_2\notin M_1$ and $x_1,x_2\notin M_2$, which seems like a dead end. What's the right way to do this? prove or disprove: For a prime number p, there is a positive integer n such that $p+1 $= $n^2$ if and only if $ p = 3.$ Prove or disprove: For a prime number p, there is a positive integer n such that $p+1 $= $n^2$ if and only if $ p = 3.$  first im trying to proof if p is prime number there is positive integer n such that $p+1=n^2$ implies $p=3$  suppose there is positive n integer such that $p+1=n^2$ and prove that $p=3$. $p=n^2-1=(n+1)(n-1)$ since $p$ is prime $n-1=1$,$ n = 2 $ $p+1=4$ $p=3$ hence it is proved that $p=3$  now if $p=3$ implies there is positive integer n such that $p+1=n^2 $  $p=3$ and p is prime  $p+1=n^2$, $4=n^2$, $n2$ or $n=-2$ hence it is proved that there is positive integer $n$ such that $p+1=n^2 $  is my proof correct? and also for prime number can i write $n+1=1$ instead $n-1=1$? How to tell what method was used to log in (Password vs. Keyfile) from the bashrc Can someone identify this book, where children are bred for specific jobs? Things I remember about this book:   Every child was selected for a certain job/career; If twins were born, one was always killed; Seeing colors for the first time (describing the color of an apple); Seeing war and death and poverty.   Does anyone have any idea what this book might be? I have an uploaded video and I want to cut sections of audio (strong language) or add beeps on top without reuploading the video. Why does ( exit 1 ) not exit the script? I have a script, that does not exit when I want it to.  An example script with the same error is:  #!/bin/bash  function bla() {     return 1 }  bla || ( echo '1' ; exit 1 )  echo '2'   I would assume to see the output:  :~$ ./test.sh 1 :~$   But I actually see:  :~$ ./test.sh 1 2 :~$   Does the () command chaining somehow create a scope? What is exit exiting out of, if not the script? So I am on Minecraft playing on my friends world (Not server). However, they won't go online for no reason. I want to go continue my work on that world and not start brand new on survival again. I've already made a lot of progress and would not like to lose it, can somebody help? I've already tried to get answers but they are either too complicated or for servers like IP. 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? An entire function whose real part is bounded must be constant. Is it okay to say "Yes you're." instead of "Yes you are."? Possible Duplicate:    I was having an SMS conversation with a friend and somehow &quot;Yes you're&quot; came into play in retaliation to a comment. Example:  Person 1: &quot;You are bad at English&quot;. Person 2: &quot;No I are not.&quot;. Person 1: &quot;Yes you're&quot;.  Is that acceptable? I would assume that it is. Think of &quot;don't&quot;. You can say:  &quot;No, don't.&quot;  or  &quot;I don't.&quot;  and it is a contraction like &quot;you're&quot;. So, is it okay? JavaScript adding decimal numbers issue How do I root my Nexus 7? I am wanting to gain root privileges on my Google Nexus 7 tablet, running Android 4.1-4.2.1. How can I do this? Inserting a term into a custom taxonomy function $f(x)=x^{-(1/3)}$ , please check whether my solution is correct? Let $f(x)=x^{-(1/3)}$ and $A$ denote the area of region bounded by $f(x)$ and the X-axis, when $x$ varies from -1 to 1. Which of the following statements is/are TRUE?   f is continuous in [-1, 1] f is not bounded in [-1, 1] A is nonzero and finite     I try to explain :    graph is  :     False , since left limit is not equivalent to right limit. True , since  f(x) rises to $infinite$ at $x=0$. True , since we can calculate it by doing integrating the function