The definition of $e$ is:  $$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$  If we use the Binomial Theorem on the function itself:  $$\left(1 + \frac{1}{n}\right)^n = \sum_{k=0}^{n} \binom{n}{k}\frac{1}{n^k} = \sum_{k=0}^{n} \frac{n!}{k!(n-k)! \cdot n^k}$$  At this point I am not sure where to go, how to both manipulate this expression and also have it be a valid input to the limit definition. Is there a simple approach that does not require Calc II knowledge, or is this one of those problems where there's really just one way to do it that everyone uses?
How do you prove that $e=\sum_{n=0}^{\infty}\frac{1}{n!}$? Here I am assuming $e:=\lim_{n\to\infty}(1+\frac{1}{n})^n$. Do you have any good PDF file or booklet available online on this? I do not like how my analysis text handles this...
Is sudo required for installing VMware Tools? I'm trying to install VMware Tools in Ubuntu 16.04. Should I do this by using sudo? Why? How to add a modifier to a collection? My goal is to reduce the faces of an object using the decimate modifier. The object I'm editing has lots of vertex groups. Blender 2.8 only lets me add the decimate modifier to a single group at once which would take quite some time. I have already put all groups in a single collection.  How can I add the decimate modifier to the entire collection?  Do I have to add the modifier group by group? Find the radius of the inscribed circle of a right triangle. Find the radius of the inscribed circle of a right triangle. The triangle's height is $\sqrt{6} + \sqrt{2}$ while the bisector of the right angle is 4.  This seemed like a generic similar triangles problem but it isn't that simple when I tried to solve it. Any help?  EDIT: Here is a picture of the problem When using processing tool in pyqgis and run, it always creates a progress bar. Is there a way to disable it?  processing.runalg('qgis:fixeddistancebuffer', _layer, distance_tolerance, 10, False, None)   I want to disable it because I get this error when I run my script in travis:     wrapped C/C++ object of type QProgressBar has been deleted See log for   more details       RuntimeError: wrapped C/C++ object of type QProgressBar has been   deleted Why use a ReentrantLock if one can use synchronized(this)? How to save user-entered line breaks from a TextArea to a database? How do I save user-entered line breaks from a &lt;textarea&gt; HTML element to a database?  It always removes the line breaks. How does ServerName and ServerAlias work? It's the following part of a virtual host config that I need further clarification on:  &lt;VirtualHost *:80&gt;   # Admin email, Server Name (domain name), and any aliases   ServerAdmin example@example.com   ServerName  141.29.495.999   ServerAlias example.com ...   This is and example config, similar to what I currently have (I don't have a domain name at the moment).  &lt;VirtualHost *:80&gt; - Allow the following settings for all HTTP requests made on port 80 to IPs that this server can be contacted on. For instance, if the server could be accessed on more than one IP, you could restrict this directive to just one instead of both.  ServerName - If the host part of the HTTP request matches this name, then allow the request. Normally this would be a domain name that maps to an IP, but in this case the HTTP request host must match this IP.  ServerAlias - Alternate names accepted by the server.  The confusing part for me is, in the above scenario, if I set ServerAlias mytestname.com and then made an HTTP request to mytestname.com, there would have to be a DNS record pointing to the server's IP for this to work? In which case, is ServerAlias just basically EXTRA ServerName entries?  Say I had a DNS entry such that foobar.com = 141.29.495.999 but then I had ServerName = 141.29.495.999 and ServerAlias was empty, would that mean that although foobar.com gets resolved to the right IP, because there is no reference to accept foobar.com in ServerName or ServerAlias?  Or something. Man I'm confused. How to do "Files to store without compression" in 7-zip? Is there any way to selectively prevent certain file types from compression in 7-zip, like in WinRAR?  I'm trying to compress a huge folder (many gigabytes), and I don't want the already-compressed files (e.g. ZIP files) to be re-compressed -- they should be stored without compression. Let $A$ be a closed and bounded subset of $\mathbb{R}^{n}$. Given that $f: A \to A$ satisfies $d(f(x),f(y)) &lt; d(x,y)$ for all $x,y \in A$, $x\neq y$ where $d$ is the usual Euclidean metric. Show that $f$ has a unique fixed point.    My observations:    Since $\mathbb{R^{n}}$ is complete, then $(A,d)$ is complete since $A$ is closed; Since $A$ is bounded, then $A \subset B_{R}$ for some $R &gt; 0$; Banach Fixed Point Theorem guarantees a unique fixed point if $f$ is contraction on $A$.   Please correct me if I am wrong. My quandry here is showing that $f$ is a contraction, i.e.  $$ d(f(x),f(y)) \leqslant k\cdot d(x,y), \hspace{3mm} k \in (0,1) $$  I'm looking for some hints or tips. Thank you. I'm having trouble understanding Cantor's diagonal argument.  Specifically, I do not understand how it proves that something is "uncountable".  My understanding of the argument is that it takes the following form (modified slightly from the  article, assuming base 2, where the numbers must be from the set $ \lbrace 0,1 \rbrace $):  \begin{align} s_1 &amp;= (\mathbf{0},1,0,\dots)\\ s_2 &amp;= (1,\mathbf{1},0,\dots)\\ s_3 &amp;= (0,0,\mathbf{1},\dots)\\ \vdots &amp;= (s_n \text{ continues}) \end{align}  In this case, the diagonal number is the bold diagonal numbers $(0, 1, 1)$, which when "flipped" is $(1,0,0)$, neither of which is $s_1$, $s_2$, or $s_3$.  My question, or misunderstanding, is: When there exists the possibility that more $s_n$ exist, as is the case in the example above, how does this "prove" anything?  For example:  \begin{align} s_0 &amp;= (1,0,0,\mathbf{0},\dots)\ \ \textrm{ (...the wikipedia flipped diagonal)}\\ s_1 &amp;= (\mathbf{0},1,0,\dots)\\ s_2 &amp;= (1,\mathbf{1},0,\dots)\\ s_3 &amp;= (0,0,\mathbf{1},\dots)\\ s_4 &amp;= (0,1,1,\mathbf{1},\dots)\\ s_4 &amp;= (1,0,0,\mathbf{1},\dots)\ \ \textrm{ (...alternate, flipped } s_4\textrm{)}\\ s_5 &amp;= (1,0,0,0,\dots)\\ s_6 &amp;= (1,0,0,1,\dots)\\ \vdots &amp;= (s_n \text{ continues}) \end{align}  In other words, as long as there is a $\dots \text{ continues}$ at the end, the very next number could be the "impossible diagonal number", with the caveat that it's not strictly identical to the "impossible diagonal number" as the wikipedia article defines it:     For each $m$ and $n$ let $s_{n,m}$ be the $m^{th}$ element of the $n^{th}$ sequence on the list; so for each $n$,      $$s_n = (s_{n,1}, s_{n,2}, s_{n,3}, s_{n,4}, \dots).$$      ...snip...      Otherwise, it would be possible by the above process to construct a sequence $s_0$ which would both be in $T$ (because it is a sequence of 0s and 1s which is by the definition of $T$ in $T$) and at the same time not in $T$ (because we can deliberately construct it not to be in the list). $T$, containing all such sequences, must contain $s_0$, which is just such a sequence. But since $s_0$ does not appear anywhere on the list, $T$ cannot contain $s_0$.      Therefore $T$ cannot be placed in one-to-one correspondence with the natural numbers. In other words, it is uncountable.   I'm not sure this definition is correct, because if we assume that $m = (1, \dots)$, then this definition says that "$s_n$ is equal to itself"&mdadsh;there is no "diagonalization" in this particular description of the argument, nor does it incorporate the "flipping" part of the argument, never mind the fact that we have very clearly constructed just such an impossible $T$ list above.  An attempt to correct the "diagonalization" and "flipping" problem:  $$s_n = (\lnot s_{m,m}, \lnot s_{m,m}, \dots) \quad \text{where $m$ is the element index and} \quad\begin{equation}\lnot s_{m,m} = \begin{cases}0 &amp; \mathrm{if\ } s_{m,m} = 1\\1 &amp; \mathrm{if\ } s_{m,m} = 0\end{cases}\end{equation}$$  This definition doesn't quite work either, as we immediately run in to problems with just $s_1 = (0),$ which is impossible because by definition $s_1$ must be $ = (1)$ if $s_1 = (0)$, which would also be impossible because...   Or more generally, with the revised definition there is a contradiction whenever $n = m$, which would seem to invalidate the revised formulation of the argument / proof.  Nothing about this argument / proof makes any sense to me, nor why it only applies to real numbers and makes them "uncountable". As near as I can tell it would seem to apply equal well to natural numbers, which are "countable".  What am I missing? Solutions to $a+b+c=12$, $a, b, c \in \mathbb{N}_0$ Let $a, b, c \in \mathbb{N}_0$. If $a+b+c=12$, how many solutions $(a,b,c)$ satisfy the equation?   Is the answer:      $729$ Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ have the same number of elements for each order, but $G$ is not isomorphic to $H$? I suspect that there is, but I haven't thought of one. Let $f:S^{2n} \rightarrow S^{2n}$ be a continuous map. Show that          there exists $x \in S^{2n}$, such that $f(x) =x$ or $f(x) = -x$;   any continous map $g: \mathbb R P^{2n} \rightarrow \mathbb RP^{2n}$ has a fixed point;      I think 1 implies 2. So, how can I find an $x$ in 1?  Thanks for the help. 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? Can I disable pinch to show all open tabs in Safari? What is the difference between Ubuntu and its derivatives? What different Ubuntu derivatives are there? And what is special with them? How to get correct scale values? The scale of my map is wrong, no matter what coordinate system I use in my project. The map and all objects show the correct coordinates but the scale of the map shows nonsense and the measurement tool gives wrong dimension.   The map should be like 1:10000 but qgis shows 1:1. Anyone an idea? How do I remove objects from an array in Java? Given an array of n Objects, let's say it is an array of strings, and it has the following values:  foo[0] = "a"; foo[1] = "cc"; foo[2] = "a"; foo[3] = "dd";   What do I have to do to delete/remove all the strings/objects equal to "a" in the array? While using the mobile site on my Android phone, I saw a comment that I wanted to flag but there was no flag button. After searching, I saw  that says flagging comments is not supported on mobile site.  Could the ability to flag comments be added to the mobile site? Do spells with a range of "Self" target the caster? New to D&amp;D, and thought about creating a Beast Master. I was wondering if spells with a range of "Self" would count you as the "target" of the spell. There's some debate in my group, as unlike something such as Cure Wounds, which specifies "A creature you touch," there's technically no mention of a target in a Self range spell within the description. A fair amount of people in my playgroup come from a TCG background, where the presence (or lack thereof) of keywords like "target" are essential for determining interaction.  tl;dr Would Paladin Smites or Thunderwave count as "targeting" yourself, and therefore be affected by Beast Master's Share Spell?