This is analogous to , but I need a concrete explanation of the induction part.   Prove that $$ \sum_{n=0}^{\infty} {n+k \choose k}x^n = \frac{1}{(1-x)^{k+1}} $$ for all $k \in \mathbb{N}$ and all $x \in (-1,1)$.  My attempt:  When $k=0$, this reduces to geometric series: $\sum_{0}^{\infty}x^n = \frac{1}{1-x}$.  When $k \geq 1$, suppose $S(k) = \sum_{n=0}^{\infty} {n+k \choose k}x^n = \frac{1}{(1-x)^{k+1}}$, how can I show that it holds for $S(k+1)$?  In particular, how can I show that $S(k+1) = S(k)+xS(k+1)$?
I recently found a formula to help find a sequence given a generating function:  $(\frac{1}{1-x})^k=\sum_{n=0}^{\infty}\binom{n+k-1}{n}x^n, k \in \mathbb{N}$  I just wondered if there was a known proof for this, or if anyone knows of a way to prove it.
Why does centering not work with calligra? Extended Euclidean Algorithm: backward vs. forward I'm confused about how to do the extended algorithm. For example, here's the gcd part gcd(8000,7001)  $$\begin{align}8000 &amp;= 7001\cdot1 + 999\\ 7001&amp;=999\cdot 7+8\\ 999&amp;=8\cdot 124+7\\ 8&amp;=7\cdot 1+1\\ 7&amp;=1\cdot 7+0\\ \gcd(8000,7001)&amp;=1\end{align}$$    Now, the extended algorithm  $$\begin{align}1 &amp;= 8 - 1\cdot7\\ &amp;= 8 - 1\cdot(999 - 8\cdot124)\\ &amp;= -1\cdot999 + 8\cdot125\end{align}$$  How do you get this 8*125 from the previous line? RAID10 vs. RAID5 for data files We are trying to evaluate the potential performance improvements that could be gained by switching an existing applications .mdf files to a RAID10 LUN vs. the existing RAID5 LUN they are living on right now. The transaction log already lives on a RAID10 LUN.   So the question is, if we move data files to RAID10 would the end user see performance improvements in the GUI? or would it be just faster writes to the data files when a checkpoint occurs and the changes get written to the data file but no improvement would be seen at the end user level? Alternate exits in Bit Trip Runner 2 Is there any point in taking the alternate exits in Bit Trip Runner 2?  I can't see any benefit to doing so in terms of unlocking a different path in the overworld, so is it just for your own benefit? Writing string at the same position using Console.Write in C# 2.0 Make clear which profile data is visible to whom For an architectural visualization, I would like to use MakeHuman models, but:   Only get them to render white with a black outline (on reflection surface too, if possible). Interact with light like normal human bodies in the scene (cast shadows, etc).   I can do the outline in freestyle, but any in-Cycles solution would be appreciated (OSL?) Let $SL_n(\mathbb{R})$ be the set of $n\times n$ matrix with determinant one. Where can I find a proof that $[SL_n(\mathbb{R}),SL_n(\mathbb{R})]=SL_n(\mathbb{R})$? I've seen this result mentioned in several places, but couldn't find a proof.  For reference, the commutator $[SL_n(\mathbb{R}),SL_n(\mathbb{R})]$ is defined as the group generated by the set of all elements of the form $a^{-1}b^{-1}ab$, with $a,b\in SL_n(\mathbb{R})$. This is a 2015 IMO problem.  It seems difficult to solve.     Find all triples of positive integers $(a, b, c)$ such that each of the numbers $ab-c$, $bc-a$, and $ca-b$ is a power of $2$.   Four such triples are $(a,b,c)=(2,2,2),(2,3,2),(3,5,7),(2,6,11)$. Does iCloud Drive automatically download all files? I am using OS X 10.10 with iCloud Drive. I am thinking of upgrading to a bigger storage plan (at the moment: 5GB).   I am running this Version on a MacBook Air with 128GB.      If I buy the 500GB package, does it mean that I fill up my local hard   drive automatically when I upload things to the iCloud Drive?   iTunes has the Option of stream the music i didn't download yet.   How does iCloud Drive behave? Is there any Online Document? Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$ $x,y,z &gt;0$, prove   $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$   Note: Often Stack Exchange asked to show some work before answering the question. This inequality was used as a proposal problem for National TST of an Asian country a few years back. However, upon receiving the official solution, the committee decided to drop this problem immediately. They don't believe that any students can solve this problem in 3 hour time frame.   Update 1: In this forum, somebody said  that BW is the only solution for this problem, which to the best of my knowledge is wrong. This problem is listed as "coffin problems" in my country. The official solution is very elementary and elegant.   Update 2: Although there are some solutions (or partial solution) based on numerical method, I am more interested in the approach with "pencil and papers." I think the approach by  in  may help.    Update 3: Michael has tried to apply Peter Scholze's method but not found the solution yet.  Update 4: Symbolic expanding with computer is employed and verify the inequality. However, detail solution that not involved computer has not been found. Whoever can solve this inequality using high school math knowledge will be considered as the "King of Inequality". Korea: Visa required to go to the city while in transit in Seoul for a Philippine citizen? Where can I get an Ubuntu Installation CD/DVD/USB? I need to upgrade my existing Ubuntu system, and I also want to install Ubuntu on a new laptop.  I would like to have an installation CD or DVD to do both of these.  Where exactly I can order or request an Ubuntu installation CD/DVD/USB? How can I keep low resolution textures pixelated? I have a problem. I am going to make a minecraft animation in blender. But when I put a minecraft texture on a 3d cube that is 16x16x16 it is very low resolution is there a way to get the texture in a better resolution or is there a way to make blender make the texture better?  I hope you have an answer. Thanks in advance Why does speed of light have to be constant? My question is not about why speed of light has a particular constant value which has been sufficiently addressed in other questions on SE already. I want to know whether the fact that speed of light has to be constant is a consequence of other fundamental laws of Universe, or is it just a fundamental law itself.  My motivation for asking this question is thinking about Special Theory of Relativity. One way of understanding why time dilation and length contraction happen is to look at it from the perspective of constancy of speed of light. Alice who is on the moving source of light and Bob, a fixed observer see the speed of light as constant. In order for Alice to see the speed of light as a constant value, time slows down and length expands (in forward direction) thus keeping the speed constant.   In the  I'm doing, the Professor tells that this constancy of speed of light could be more generally referred to as, "Principle of wave constancy". That is, the speed of any wave is a property of the medium (and thus it remains constant). So, the speed of a wave in water, for example, remains constant even if the source of the waves (say a paddle keeps moving). Frequency and wavelength may change but the speed remains constant. In a similar fashion, light is an electromagnetic wave and thus its speed is constant. And the constant speed is defined by the medium (which happens to be vaccum i.e., no medium for light).   My questions are twofold now -    Is the above reasoning correct to explain why speed of light is constant? (i.e., thinking of it as constancy of any wave in general) Theoretically, could there be other waves that could travel through vaccum at a different speed? If yes, time dilation and length contraction will have to be different.   I'd appreciate it if the answer is kept non-technical (essential math is okay). More importantly, I request you to try and explain without using General Theory of Relativity. Because I'm thinking how the scientific understanding by 1905 had already knew that speed of light has to be constant. Java: Difference between the setPreferredSize() and setSize() methods in components Show that $(1+\frac{x}{n})^n \rightarrow e^x$ uniformly on any bounded interval of the real line. Show that $(1+\frac{x}{n})^n \rightarrow e^x$ uniformly on any bounded interval of the real line.   I am trying to argue from the definition of uniform convergence for a sequence of real-valued functions, but am struggling quite a lot. My efforts so far have concentrated on trying to find a sequences, ${a_n}$ which tends to zero, such that   $$|(1+\frac{x}{n})^n -e^x |\leq a_n$$ for all $n$. But I have been unsuccessful thus far. All help is greatly appreciated. Is it: My apples and orange are/is wrong? Simple question:     My apples and orange are wrong   or     My apples and orange is wrong   I am not a native English speaker, and I am having some trouble choosing between plural are or singular is for that kind of example. Find all entire functions $f(z)$ such that $|f(z)|=1$ for $|z|=1$ Show $f$ is analytic if $f^8$ is analytic This is from Gamelin's book on Complex Analysis.  Problem: Show that if $f(z)$ is continuous on a domain $D$, and if $f(z)^8$ is analytic on $D$, then $f(z)$ is analytic on $D$. (I assume the intention is that $D$ is nice, i.e. open and connected)  I'm not exactly sure how to approach this. I'm guessing it has something to do with zeroes of $f$. For example, around non-zeroes, $f$ is analytic since $z^{1/8}$ is nonzero in a neighborhood. Or something along those lines. The details are eluding me.  Any help would be appreciated!