I already known how to prove that $\lim_{x\rightarrow 0}x^{x^x}=0$ and $\lim_{x\rightarrow 0}x^x=1$. I also tried to use L'Hôpital's rule for this question but it didn't work. How to find the limit? (The limit should be $1$ from the graph sketching.)
I'm a computer student, learning math just for fun. Today I was  that I found something strange! I noticed that that wired function ${x^{x^{\cdot^{\cdot^{x}}}}}$ in zero, seems to converge to 1 when there are even powers and to 0 when there are odd powers! Then I attempt to prove it but I failed.  Then I did a little research and I found the  article on Wikipedia. This article says that my guess was right but without any proof. So I'm here to ask you about it.     If we define ${x^{x^{\cdot^{\cdot^{x}}}}}$ as ${^{n}x} :=\begin{cases} 1 &amp;\text{if }n=0 \\ x^{\left(^{(n-1)}x\right)} &amp;\text{if }n&gt;0 \end{cases}$      then prove:      $$\lim_{x\rightarrow0} {}^{n}x = \begin{cases}   1, &amp; n \text{ even} \\  0, &amp; n \text{ odd} \end{cases}$$
The center of the dihedral group How to prove that the center of the dihedral group $D_{2n}$ is $\{1,r^{n}\}$ and the center of $D_{2n-1}$ is $\{1\}$?  I don't know how to prove it in this general case. Does a wireless-N (802.11n) network have poor performance when in b/g "mixed" mode? It seems to be an "old" wives tale that, when using a wireless-N router in "mixed mode" to support legacy 802.11b or 802.11g devices, the performance of 802.11n clients will suffer.      claim that when running in mixed-mode, all (some?) N clients run at G speeds.   make the same claim, but say it only happens when a G client is connected.    say N clients run faster, but still run about 30% slower than if the router were in N-only mode, even if there are no legacy B/G clients connected.    claim there is no speed drop for N clients when running on a mixed-mode network.  They say the only issue is that overall network throughput will be lower, because only one client can be transmitting at any time, so some of that transmission time must be shared with the legacy B/G clients running at lower speeds, reducing the overall throughput from what it would be if there were only N clients connected.    So, which is it?  Will running in mixed-mode slow down my network, even if there are no B/G clients?  If I'm running N, will having another client connected at B/G slow me down substantially vs. if they were running N? Grandi's Series; tends to $1/2$, but why is this considered a valid sum? Grandi's series,  $$1+1-1+1-1+1-1+...$$  can be expressed as the below:  $$\sum_{n=0}^\infty(-1)^n$$  Two valid sums that make sense to me are $1$, and $0$, depending on how you approach the series. $(1+1)-(1+1)-(1+1)-...=0$, and $1+(1-1)+(1-1)+(1-1)+...=1$.  There is consensus, however, that the actual sum is $\frac{1}{2}$. Why? I understand the approach of finding partial means of the series, and they do indeed tend to $\frac{1}{2}$, but it seems unintuitive to assert that the sum is neither $1$ or $0$.  A more convincing method I found was assuming the series is $S$, then shifting it such that $S-1 = S$, then through algebra finding $S = \frac{1}{2}$, but again, it seems more intuitive answer is either $0$ or $1$. I say this strictly because adding and subtracting integers should equal an integer, never a fraction.  Is this a characteristic of infinite series, which is not specific to Grandi's series? How can I correct this code to get an answer? While looking over various PHP libraries I've noticed that a lot of people choose to prefix some class methods with a single underscore, such as  public function _foo()   ...instead of...  public function foo()   I realize that ultimately this comes down to personal preference, but I was wondering if anyone had some insight into where this habit comes from.  My thought is that it's probably being carried over from PHP 4, before class methods could be marked as protected or private, as a way of implying "do not call this method from outside the class".  However, it also occurred to me that maybe it originates somewhere (a language) I'm not familiar with or that there may be good reasoning behind it that I would benefit from knowing.  Any thoughts, insights and/or opinions would be appreciated. What does “” exactly mean in “echo  temp.txt”? rewrite access to collection to avoid "double" finding How to archive the whole website? To save a single page via Wayback Machine I can go to:      How can I archive the website recursively by Wayback Machine?    There is some project such as , but I'm looking for a feature which allows me the upload the website recursively. Prove every element of $G$ has finite order. Let $G$ be a group such that the intersection of all its subgroups which are different from $\{e\}$, is a subgroup different from $\{e\}$. Prove that every element of  $G$  has finite order.  Assume that $a$ has infinite order in $G$. Then let $y\neq e$ belong to the intersection of all subgroups of $G$ different from $\{e\}$. Then $y=a^n$ and $y$ will also belong to subgroup generted by $a^2$ hence  $y=a^{2m}$ for some integers $m$. This contradicts that $a$ has infinte order.  Is this correct? Sum of nilpotent and element is a unit in ring? I can't tell where I went wrong, am still a beginner.    Let N be a nilpotent in a commutive ring , let X be any element.   I'll will be showing X + N is a unit .   Assume Y(X+N) =1   Then YX(N^n-1) =  N^n-1   Multiplying both side by  X+N   We get X(N^n-1) =  X(N^n-1)  Thus nilpotent summed with any element gives unit. Earlier today, I was suspended from chat due to an inappropriate message (I totally deserved it, but that's not the point). I looked at my phone, and this is all that it told me:    ...which confused me because chat was still moving as normal. I didn't realize that I had been suspended until I checked on my computer to see what the heck was going on.  To me, "This room is read-only" sounds like some sort of database lockup or something related specifically to the room, not the user. It doesn't give any feedback to indicate that the room will be "read-only" for 30 minutes, and doesn't mention that all other rooms are also "read-only". Even a message saying just "You have been suspended from chat" would be more helpful. I am having trouble proving the statement:     Let $$S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$$ Prove that for every $\epsilon &gt; 0$, the intersection of $S$ and $(0, \epsilon)$ is nonempty. Does "a couple" always mean two? During my upcoming holiday to the USA I'm thinking about buying a new guitar as the exchange rate is extremely favorable. Excuse my naivety but I have never flown with a guitar before. What is a recommended way to go about this? Could I take it as carry on luggage or would it have to be checked? If checked luggage what kind of precautions should I take to keep it safe? Are there any problems with flying US budget carriers? If I do buy the guitar I'm expecting to have to make two flights with it, one domestic flight with Virgin America and an international flight with Air NZ. How can I produce the codes to get the symbols of pencil, ruler and compass in LATEX? I'm gonna write a sentence I can draw this figure by using simple tools in my presentation. So I prefer to include symbols pencil, ruler and compass instead of the words simple tools. Is it possible? Forge CBC-MAC by creating a new message, given the MAC of two messages Prove $(a+b)^\frac{1}{n} \le a^\frac{1}{n} + b^\frac{1}{n}$ Prove: $$ (a+b)^\frac{1}{n} \le a^\frac{1}{n} + b^\frac{1}{n}, \qquad \forall n \in \mathbb{N} $$ I have have tried using the triangle inequlity $ |a + b| \le |a| + |b| $,  without any success. Is it legal to add your publications to ResearchGate? I recently created a profile on . And to add your publications list it requires you to upload the paper. Is it legal to upload published journal articles on public servers, where everyone can download it for free? Why is uniform prior on log(x) equal to 1/x prior on x? I'm trying to understand Jeffreys prior. One application is for 'scale' variables like the standard deviation $\sigma$ (or its square, the variance $\sigma^2$) of Gaussian distributions. It is often said that using a uniform prior over $\sigma$ is not really non-informative and instead one should either:   Use instead $\ln \sigma$ as the free parameter, with a uniform prior (this is often called a log-uniform prior) Or keep using $\sigma$ as the free parameter but use $1/\sigma$ as the prior (which is not uniform).   Why are the above two methods/priors equivalent? I feel it has something to do with the fact that the derivative of ln $\sigma$ is $1/\sigma$ but I can't take the next step.   Also, why does this even matter, in simple language with minimal jargon? I see all these complicated explanations online involving the Fisher information matrix but in the end all I see is that the above log-uniform or $1/\sigma$ priors preferentially weight lower values of $\sigma$ more highly. Why? If possible, a simple analytic example or python snippet would be very helpful. Database synchronization between dev/staging and production