Assuming  $$e^{x}=\sum_{i=0}^{\infty}\frac{x^i}{i!},$$  show that $e^{x}e^{y}$ = $e^{x+y}$.  I have expanded both but am not sure if I need to multiply the two summations and show that product or if there is a simpler way of showing this.
In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following:   $$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over n!}\Big)=\sum_{n=0}^{\infty}\sum_{k=0}^n{x^ky^n \over {k!n!}}$$  But, where should I go next to get $e^{x+y}=\sum_{n=0}^{\infty}{(x+y)^n \over n!}$.  Thanks in advance.
I've done some research, and found this on Wikipedia. \begin{matrix}a\uparrow b=a^{b}=&amp;\underbrace {a\times a\times \dots \times a} \\&amp;b{\mbox{ copies of }}a\end{matrix} \begin{matrix}a\uparrow \uparrow b&amp;={\ ^{b}a}=&amp;\underbrace {a^{a^{{}^{.\,^{.\,^{.\,^{a}}}}}}} &amp;=&amp;\underbrace {a\uparrow (a\uparrow (\dots \uparrow a))} \\&amp;&amp;b{\mbox{ copies of }}a&amp;&amp;b{\mbox{ copies of }}a\end{matrix} I'm still a bit confused on this topic. I've done a lot of research, but I can't seem to understand this. Could somebody explain this in an understandable way? From what I can gather, for example 2↑↑3 would be 2^2^2 = 16? Then 2↑↑↑3 would be 2^2^2^2 = 256? Am I correct, or have I mis-interpreted this? Finding the dual basis for the space of polynomials of degree $\leq$ 2 Let $V$ be the space of polynomials of degree at most $2$, with basis $\{1,x,x^2 \}$.Find the dual basis to $\{1,x,x^2\}$.   I know that the dual basis of $\{f_1,f_2,f_3\}$ is the basis $\{f_1^*,f_2^*,f_3^*\}$ where $f_i^*(a_1f_1,\ldots,a_3f_3)$ but I'm not sure how to actually compute the dual basis. Samsung galaxy S4 Blacked out Screen need to retreive files The passive with "let" In Minecraft, is there any way to make tools indestructible (without using mods)? I want to make some tools indestructible (for an adventure map). Is it possible to do this using enchantments (or a command of some kind?) 8 bits representing the number 7 look like this:  00000111   Three bits are set.     What are algorithms to determine the number of set bits in a 32-bit integer? I have to format my entire disk and reinstall Ubuntu.  I have installed a lot of software on my current system. And I will have to reinstall all those updates, drivers and applications too.  Is there any way I can make an .iso image of my installed Ubuntu along with all the software, preferences, drivers etc in it ?  Is there any way I can back up my whole OS along with all its software, drives, updates and all ? Linux command line. Move all files and directories in directory, except some files and directories I have a folder A which has files and directories, I want to move all those files and directories to another folder B, except file, file2, directory, and directory2.  How can this be done? Why Are Ethernet/RJ45 Sockets Magnetically Coupled? As the title says really, why do ethernet sockets need to be mag-coupled? I have a basic understanding of electronics, but mostly, I can't figure out the right search terms to google this properly. In convex optimization, the feasible region is convex if equality constraints $h(x)$ are linear or affine, and inequality constraints $g(x) \leq 0$ are convex. Does this mean that if $h(x)$ is nonlinear, then the optimization problem is non-convex? Is there any special case where $h(x)$ is nonlinear and the feasible region is still convex? How to make a redstone clock that I can turn on/off with a lever? I want to create a Redstone clock that can be enabled and disabled with a lever permanently placed down. I'm using Mathematicato produce some of my graphics (when i'm not able to just TikZ them easily). All these images are availabvle as .eps and .pnghaving the same file name - let's use img.eps / img.png in the following.  I would like to set a preference for graphicx to use only one type, i.e. using  \includegraphics[width=.6\textwitdh]{img}   I would like to specify, which img to use with one switch, (package) option or variable.  I tried  \DeclareGraphicsExtension{.eps}   to restrict the allowed image extension to .eps, but still the .png file is included. It seems, .pngis preferred over .eps.  Though this is of course restricted to XeTeX it might be interesting for others, if you use .png together with some lossy format like .jpg to have a file extension preference.   Is there a package option for that or an easy command (global document option) for that?  To provide an MWE is - I think - difficult, because it would be necessary to provide at least 3 files. sort() and reverse() functions do not work When given $p = 5, q = 11, N = 55$ and $e = 17$, I'm trying to compute the RSA private key $d$.  I can calculate $\varphi(N) = 40$, but my lecturer then says to use the extended Euclidean algorithm to compute $d$. That's where I get stuck.    Here's my work so far:  First I use the Euclid algorithm to calculate:  40 = 2(17) + 6  17 = 2(6) + 5 6 = 1(5) + 1 = gcd   So I know the GCD is 1. Applying the 'extended' section of the algorithm:  6 = 40-2(17) 5 = 17-2(6) 1 = 6-1(5) 1 = 6-1(17-2(6)) 3(6) = 1 (17) 3(40 - 2(17)) - 1(17) 3(40) - 3(17)   I know the answer is $33$, but I have no idea how to get there using the extended Euclidean algorithm. I can't figure out why I'm getting $3(40) - 3(17)$ when I know the answer should contain $33$. Will wolves attack Ocelots? Will wolves attack and kill ocelots in Minecraft? I know they kill sheep. I just don't want my cats being eaten. How to change the default fonts in Windows 10 I have read the entries for other OS versions.    Changing the MS Shell Dlg (2) entries in FontSubstitues registry key doesn't seem to work (in Windows 10), or it works in two minor places.  I don't have access to a Windows 95/98/etc. machine, the "earliest" I have is Windows 8.1, so I can't use the Plus directory trick either (directory doesn't exist). For the WindowMetrics registry entry, I seem to have all the *Font keys to be set to the font I want, but that doesn't seem to be doing the trick (on Windows 10). For Windows 8/8.1, there was a program called Windows 8 Font Changer, which worked like a charm, but doesn't seem to on Windows 10 (though it was probably that program run that changed the fonts in the WindowMetrics).   Thanks.   Update: One more method that came out that is perhaps still not the solution: Winaero Tweaker (winaero.com), as it has some issues, the biggest two of which seem to be:   If you have changed the size of text etc. from: "Settings –> Display –> Advanced display settings –> Advanced sizing of text and other items –> set a custom scaling level (not recommended)", the changes do not stick upon a logoff-login or restart. You need to have changed it, if at all, only through "Settings –> Display –> Change the size of text, etc." (which goes up to only a certain percentage, depending on the maximum resolution of your device – 125% on a tablet of mine, which I was able to set to 175% or even 200% with the other, older method that is now apparently listed as not recommended).  Even though it lists a number of items to change the font and size of, not everything gets changed. A typical example of this is the right click menu on the desktop, which still shows the View, Sort By, etc. items in the default font, however much it has a clear 'Change menu font' item. How do I get an operating system onto a new hard drive? Types of circular acceleration? To my knowledge there are three types of acceleration when a body (e.g. a rod) is moving in a circle about an axis. These are:   Angular acceleration : this is the rate of change of angular velocity. Tangential acceleration : this is the linear acceleration of the system in a tangential direction to the circle and equals the radius times the angular acceleration. Radial/centripetal acceleration : this is the linear acceleration of the system that is directed inwards towards the center of the circle.   I also think there are two types of velocity:   Linear velocity: this is its velocity in the tangential direction and is constantly changing Angular velocity or angular frequency: this is the rate of change of angle.   Is the above correct? And have I missed anything? I have, by default, 250MB of swap space in Ubuntu, and I want to expand it to a larger size. I need 800MB, which I think will be enough to open several applications without having to hit the current limit of swap. I hope someone can help me. If we have 2 batteries one of emf x and the other is of emf y and we connect them in series we get an effective emf of x+y.  But what if we connect them in parallel, how to calculate the emf now?