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How do i prove invertibility of this linear transformation Floating point precision while using Python's max() The Equation $11^x + 13^x + 17^x =19^x  $  Has    No Real Roots Only One Real Roots Exactly Two Real Roots  More than Two Real Roots    What I have done is   The function $f(x)=11^x + 13^x + 17^x -19^x  $ is strictly increasing and being always positive it have no roots Prove that $$1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+.........=\sqrt{3}$$   $\bf{My\; Try::}$ Using Binomial expansion of  $$(1-x)^{-n} = 1+nx+\frac{n(n+1)}{2}x^2+\frac{n(n+1)(n+2)}{6}x^3+.......$$  So we get $$nx=\frac{1}{3}$$ and $$\frac{nx(nx+x)}{2}=\frac{1}{3}\cdot \frac{3}{6}$$  We get $$\frac{1}{3}\left(\frac{1}{3}+x\right)=\frac{1}{3}\Rightarrow x=\frac{2}{3}$$  So we get $$n=\frac{1}{2}$$  So our series sum is $$(1-x)^{-n} = \left(1-\frac{2}{3}\right)^{-\frac{1}{2}} = \sqrt{3}$$  Although I know that this is the simplest proof, can we solve it any other way someting like defining $a_{n}$ and then use Telescopic sum.  Thanks. File permissions and saving Let's say you open a file on which you have write permission. Meanwhile you change permissions and remove write permission while you still have the file open in some editor.  What will happen if you edit and save it? I have the following functional equation. Find all continuous functions $f:(-1,1) \to \mathbb R$ such that $$ f(x+y)=\frac{f(x) + f(y)}{1 - f(x)f(y)} $$ The first obvious solution is $f(x) \equiv 0$. Another one I guessed, it is $f(x) = \pm \tan x$. I suspect, that there are no more solutions. The problem is that I don't know how to prove that.  Since we have a rational equation, I have no idea how to make any substitutions in order to get expression for $\tan x$ (as it is not a rational expression).  P.S. I can also show that it is true that (a) $f(0) = 0$ (take $x=y=0$), and (b) $f(-x) = -f(x)$ (take $y=-x$).  Update: there is a solution of this problem also here: Finding an unknown linear transformation given that $T(1,1)=(1,0,2)$ and $T(2,3) = (1,-1,4)$ Let $T\colon\mathbb{R}^2\to\mathbb{R}^3$ be the linear transformation such that $T(1,1)=(1,0,2)$ and $T(2,3) = (1,-1,4)$.       Does such a linear transformation exist?  So far I've worked out that it cannot exist, as the first entry for $T(1,1)$ is 1, whereas the first entry for $T(2,3)$ is 1, which can only be obtained through 3-2, or half of 2.  Is there any way of presenting this formally? Let $a,b,c>0$ such that $a+b+c=3$, find max/min of $\frac1{a^2+1}+\frac1{b^2+1}+\frac1{c^2+1}$ Let $a,b,c&gt;0$ be real numbers such that $a+b+c=3$, find the maximum and minimum of:   $$\frac1{a^2+1}+\frac1{b^2+1}+\frac1{c^2+1}$$   I've tried using AM-GM-HM, but I can't find the maximum or the minimum.  Using intuition, substituting $a=b=c=1$ should give us one of the optimums, but I cannot prove that it is the optimum. If my Schengen visa starts on the day of arrival which is one day ahead of departure, would they let me get into the plane? Android How to use MediaScannerConnection scanFile I'm very new to presentations in LaTeX (I've been using PowerPoint for years but have finally converted). I am wondering how to get a GIF into a section. If need be, I can convert a GIF into lots of pictures and have them overlay to give the impression of movement. I have a motherboard with a PCIexpress x16 slot.  I want to purchase a card called AMD Radeon HD 7770, which uses PCIe version 3.0.  Is that backwards compatible? Related questions in "Ask Question" page is better than the search tool Maximum length of tags Possible Duplicate:        Is there a maximum limit on the length of tags? I noticed that the tag for SQL Server Management Studio is truncated to sql-server-management-stu.  It seems like this should be an easy thing to correct, if not to allow the creation of new longer tags, but to correct existing tags that have been truncated. How to reduce the size of VMware disk My Ubuntu image VMware disk has grown to 5GB physical file size. It's a dynamically growing disk with the maximum capacity of 8GB. I cleaned some stuff inside Linux distro and now inside it occupies 3GB. Obviously the physical file size didn't shrink back automatically. Is there a way (if possible free of charge) to shrink this disk back to ~3GB?  I need this to fit the image on the DVD. This is a  about RAID levels.    What are:   the RAID levels typically used (including the RAID-Z family)?  deployments are they commonly found in? benefits and pitfalls of each? iOS 7 Weather app expand/collapse transition How to view all the Symbolic links,Junction points,Hard links in a Folder using dir? The command dir /a displays a list of all the files,folders in a given location. But it displays the type  for Junction Points as well as Symbolic Links to Folders. Is there any command which will differentiate and tell me which of these  are Junction Points and which are Symbolic links as well as which of the files are hard links ? Calculating residue of pole of order $2$ Is there a good way to compute the residue of $f(z)=\dfrac{1+z}{1-\sin z}$ at $z=\pi/2$, which is a pole of order $2$?  Using the residue calculation formula yields  $$\text{Res}_{z=\pi/2}f(z)=\lim_{z\rightarrow\pi/2}\dfrac{d}{dz}\left(\left(z-\dfrac\pi2\right)^2f(z)\right)$$  The derivative is quite ugly, and calculating the limit requires L'Hospital probably twice (or more). The calculation is just too much. Is there a better way? How do you add a certificate authority (CA) to Ubuntu?