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Show $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$ How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$? I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me. (This series arises in studying the first passage time of a simple random walk.) This is from Fraleigh's First Course in Abstract Algebra (page 82, Theorem 8.16) and I keep having hard time understanding its proof. I understand only until they mention the map $\lambda_x (g) = xg$. Can someone explain this proof step by step? Thank you so much! Here is the proof:  Let $G$ be a group. We show that $G$ is isomorphic to a subgroup of $S_G$. Define a one-to-one function $\phi: G \to S_G$ such that $\phi(xy)=\phi(x) \phi(y)$ for all $x,y \in G$. For $x \in G$, let $\lambda_x: G \to G$ be defined by $\lambda_x (g) = xg$ for all $g \in G$. The equation $\lambda_x (x^{-1} c) = x(x^{-1} c) = c$ for all $c \in G$ shows that $\lambda_x$ maps $G$ onto $G$. If $\lambda_x (a) = \lambda_x (b)$, then $xa=xb$ so $a=b$ by cancellation. Thus $\lambda_x$ is also one to one, and is a permutation of $G$. We now define $\phi: G \to S_G$ by defining $\phi(x) = \lambda_x$ for all $x \in G$. To show that $\phi$ is one to one, suppose that $\phi(x) = \phi(y)$. Then $\lambda_x = \lambda_y$ as functions mapping $G$ into $G$. In particular $\lambda_x (e) = \lambda_y (e)$, so $xe=ye$ and $x=y$. Thus $\phi$ is one to one. It only remains to show that $\phi(xy) = \phi(x) \phi(y)$, that is, that $\phi_{xy} = \lambda_x \lambda_y$. Now for any $g \in G$, we have $\lambda_{xy} (g) = (xy)g$. Permutation multiplication is function composition, so $(\lambda_x \lambda_y)(g) = \lambda_x (\lambda_y (g)) = \lambda_x (yg) = x(yg)$. Thus by associativity, $\lambda_{xy} = \lambda_x \lambda_y$. Creating a dictionary from a CSV file A group such that $a^m b^m = b^m a^m$ and $a^n b^n = b^n a^n$ ($m$, $n$ coprime) is abelian? Let $(G,.)$ be a group and $m,n \in\mathbb Z$ such that $\gcd(m,n)=1$. Assume that $$ \forall a,b \in G, \,a^mb^m=b^ma^m,$$ $$\forall a,b \in G, \, a^nb^n=b^na^n.$$  Then how prove $G$ is an abelian group?  Some context: Some of these commutation relations often imply that $G$ is abelian, for example , or . This looks like another example of this phenomenon, but the same techniques do not apply. Sudden increase in timing when summing over 250 entries Is it possible to show our custom message in QGIS Status bar using python? What is the ancestral form of language used in the fable that David was studying during the flight? David studies linguistics during the flight to LV-223. We get to listen and read a fable. Is it written in a real language? What is the name of the language? Can the fable be translated?  Approximate quotation:     hjewis jasma hwaelna nahast akwunsez dad'kta, tam ghermha vagam ugenthe Removing elements with Array.map in JavaScript I would like to filter an array of items by using the map() function. Here is a code snippet:  var filteredItems = items.map(function(item) {     if( ...some condition... )     {         return item;     } });   The problem is that filtered out items still uses space in the array and I would like to completely wipe them out.  Any idea?  EDIT: Thanks, I forgot about filter(), what I wanted is actually a filter() then a map().  EDIT2: Thanks for pointing that map() and filter() are not implemented in all browsers, although my specific code was not intended to run in a browser. How do I run .sh scripts? Whenever I open a .sh file, it opens it in gedit instead of the terminal. I can't find any option similar to Right Click → Open With → Other Application... → Terminal.  How do I open this file in the terminal? What are the pros and cons of the leading Java HTML parsers? Can someone confirm that its not possible to change the height of a dropdown that is shown when you click on a select box.   The size attribute of the select makes it look like a list, the height attribute in the CSS doesnt do much good either. How do I tell if matrices are similar? How do I boot into single-user mode from GRUB? Topology: Example of a compact set but its closure not compact Can anyone gives me an example of a compact subset such that its closure is not compact please? Thank you. pgfplots: prevent single plot from being listed in legend In order to fill the area between two complex curves I use tables and the vertcat command (see ).   Now I don't want the plotted area to appear in my legend, just the two curves. Is there any command to prevent the yellow area plot from being added to my legend?  this could be a minimal example:  \documentclass{article} \usepackage{pgfplots}  \begin{document} \begin{tikzpicture}   \begin{axis}[area style,axis on top]     \addplot+[mark=none,fill=yellow,draw=none] {0.1*x^2} \closedcycle;     \addplot[mark=none,draw=red,line legend] {0.1*x^2};     \addplot[mark=none,draw=blue,line legend] {0.0*x^2};     \legend{red curve,blue curve};     % \legend{\empty,red curve,blue curve};   \end{axis} \end{tikzpicture} \end{document}   With this legend{...}-command the last plot is skipped, an additional \empty doesn't help, too.  Sadly, drawing the first to plots together with fill=yellow,draw=red is not an option for me. Does anybody know any inside hacks? As far as I know, there is no command shown in the manual. Thanks in advance. I have Xubuntu 18.04.3, and whenever I try to install a package through apt-get I get the following message.    Removing linux-image-5.0.0-31-generic (5.0.0-31.33~18.04.1) ... /etc/kernel/postrm.d/initramfs-tools: update-initramfs: Deleting /boot/initrd.img-5.0.0-31-generic /etc/kernel/postrm.d/zz-update-grub: Sourcing file `/etc/default/grub' /usr/sbin/grub-mkconfig: 35: /etc/default/grub: FROM:: not found run-parts: /etc/kernel/postrm.d/zz-update-grub exited with return code 127 dpkg: error processing package linux-image-5.0.0-31-generic (--remove):  installed linux-image-5.0.0-31-generic package post-removal script subprocess returned error exit status 1 Errors were encountered while processing:  linux-image-5.0.0-31-generic libdvd-pkg: Package libdvdcss2-1.4.2-1 was removed, stop processing... E: Sub-process /usr/bin/dpkg returned an error code (1)   Contents of /etc/default/grub:  # If you change this file, run 'update-grub' afterwards to update # /boot/grub/grub.cfg. # For full documentation of the options in this file, see: #   info -f grub -n 'Simple configuration'  GRUB_DEFAULT=0 GRUB_TIMEOUT_STYLE=hidden GRUB_TIMEOUT=10 GRUB_DISTRIBUTOR=`lsb_release -i -s 2&gt; /dev/null || echo Debian` GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" GRUB_CMDLINE_LINUX="persistent"  # Uncomment to enable BadRAM filtering, modify to suit your needs # This works with Linux (no patch required) and with any kernel that obtains # the memory map information from GRUB (GNU Mach, kernel of FreeBSD ...) #GRUB_BADRAM="0x01234567,0xfefefefe,0x89abcdef,0xefefefef"  # Uncomment to disable graphical terminal (grub-pc only) #GRUB_TERMINAL=console  # The resolution used on graphical terminal # note that you can use only modes which your graphic card supports via VBE # you can see them in real GRUB with the command `vbeinfo' #GRUB_GFXMODE=640x480  # Uncomment if you don't want GRUB to pass "root=UUID=xxx" parameter to Linux #GRUB_DISABLE_LINUX_UUID=true  # Uncomment to disable generation of recovery mode menu entries #GRUB_DISABLE_RECOVERY="true"  # Uncomment to get a beep at grub start #GRUB_INIT_TUNE="480 440 1"  FROM: GRUB_CMDLINE_LINUX_DEFAULT="" GRUB_CMDLINE_LINUX="" TO: GRUB_CMDLINE_LINUX_DEFAULT="ipv6.disable=1" GRUB_CMDLINE_LINUX="ipv6.disable=1" "There is a plethora..." or "There are a plethora..."? Consider a particle on the ground. This particle is raised by a force of magnitude $mg$ to a height $h$ above the ground. At this point, the work done on the particle by the force is $mgh$, which is equal to the potential energy of the particle. But, during this period, the force of gravity also acts on the particle and is displaced by $h$, and so does a work of $-mgh$ on the particle. Shouldn't  the two cancel and no net work should be done on the particle?   If they don't cancel, then where did the energy that came from the work done by the force of gravity go? I've been doing some practice questions in the textbook for an upcoming predicate calculus lecture and I think I've managed to get A and B (possibly C) correct but I am clueless on how I can manage D and E.   For A I got: ∃x(R(x)∧¬B)  For B I got: ∀x(¬R(x)∧B)  For C I got: ∃x(R(x)∧(x=y))   - Not sure if this is even remotely correct    R(X) : ”X is rich”  X = Y : ”X equals Y”  B : ”Bill Gates”  a) Someone besides Bill Gates is rich.  b) Bill Gates alone is rich.  c) At least two people are rich.  d) Exactly one person is rich.  e) Exactly two persons are rich.  The universe of discourse is the set of all people. $\lambda=\frac{2h}{p}$ instead of $\lambda=\frac{h}{p}$? I am studying quantum physics and there is something I don't understand:   I know that for any particle $E=hf$ (Einstein relation) and  $v=\lambda f$ ($v$ is the speed of the particle). I also know that the kinetic energy is $E_k=\frac{mv^2}{2}$.    Solving those 3 equations for $\lambda$:  $$h\frac{v}{\lambda}=\frac{mv^2}{2},$$ I finally find $$\lambda=\frac{2h}{mv}=\frac{2h}{p},$$ which is not consistent with the De Broglie relation $$\lambda=\frac{h}{p}.$$ Where am I wrong in my development?