I want to show that $$  a_n = \frac{3^n}{n!} $$ converges to zero. I tried Stirlings formulae, by it the fraction becomes $$  \frac{3^n}{\sqrt{2\pi n} (n^n/e^n)} $$ which equals $$  \frac{1}{\sqrt{2\pi n}} \left( \frac{3e}{n} \right)^n $$ from this can I conclude that it goes to zero because $\frac{3e}{n}$ and $\frac{1}{\sqrt{2\pi n}}$ approaching zero?
Why is     $$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$   Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is     $$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$     This is being repurposed in an effort to cut down on duplicates, see here: Coping with  duplicate questions.  and here: .
Possible Duplicate:          On  it's a stated fact that if $I$ and $J$ are coprime ideals of a ring $A$, then $I^k$ and $J^\ell$ are also coprime for any integers $k,\ell$.   There's no proof given. Why is this fact true? If an email message contains an email address in the body, Gmail automatically converts it into a mailto: link, which is helpful.  And many times each day I've enjoyed being able to right-click a mailto: link in an email (within Gmail) and then click "Copy email address" in the context menu.  Then I can paste that email address into a program (such as when I want to search a database).  However, within the past month or so, I've noticed that I can't right-click and click "Copy email address" without Gmail then popping open a "New Message" panel and changing my current URL to have ?compose=new at the end.  I wonder if it's related to .  How can I disable it?  I'm tried of having to click the "Discard draft" trash can button so many times throughout the day. Biblatex: submitting to a journal How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$ If $f(x)$ is a continuous function on $(-\infty, +\infty)$ and $\int_{-\infty}^{+\infty} f(x) \, dx$ exists. How can I prove that $$\int_{-\infty}^{+\infty} f(x) \, dx = \int_{-\infty}^{+\infty} f\left( x - \frac{1}{x} \right) \, dx\text{ ?}$$ I use pgfplots for most of my plots. Now I need to add a . Is this possible with pgfplots or any other package? Given a 4-vector $p^\mu$ the Lorentz group acts on it in the vector representation: $$ \tag{1} p^\mu \longrightarrow (J_V[\Lambda])^\mu_{\,\,\nu} p^\nu\equiv \Lambda^\mu_{\,\,\nu} p^\nu. $$ However, I can always represent a 4-vector $p^\mu$ using left and right handed spinor indices, writing $$ \tag{2} p_{\alpha \dot{\alpha}} \equiv \sigma^\mu_{\alpha \dot{\alpha}} p_\mu.$$ So the question is: in what representation does the Lorentz group act on $p_{\alpha \dot{\alpha}}$?    There are a lot of questions about this and related topics around physics.se, with a lot of excellent answers, so let me clear up more specifically what I am asking for.  I already know that the answer to this question is that the transformation law is  $$ \tag{3} p_{\alpha \dot{\alpha}} \rightarrow (A p A^\dagger)_{\alpha\dot{\alpha}}$$ with $A \in SL(2,\mathbb{C})$ (how is mentioned for example in ). I also understand that $$ \tag{4} \mathfrak{so}(1,3) \cong \mathfrak{sl}(2,\mathbb{C}),$$ (which is explained for example  by Edward Hughes,  by joshphysics,  by Qmechanic).  So what is missing? Not much really. Two things:   How do I obtain (3) and what is the specific form of $A$, i.e. its relation with the vector representation $\Lambda^\mu_{\,\,\nu}$? Defining the following $$ (\tilde p) \equiv p^\mu, \qquad \Lambda \equiv \Lambda^\mu_{\,\,\nu},$$ $$ \sigma \equiv  \sigma_{\alpha \dot \alpha}, \qquad \hat p \equiv p_{\alpha \dot \alpha},$$ we can rewrite (1) and (2) in matrix form as $$ \tag{5} \hat p \equiv \sigma \tilde p \rightarrow \sigma \Lambda \tilde p  = ( \sigma \Lambda \sigma^{-1}) \hat p,$$ however, this disagrees with (3) which I know to be right, so what is wrong with my reasoning? Why does the transformation law (3) has a form $$\tag{6} A \rightarrow U^{-1} A U,$$ while the usual vector transformation (1) has a form $V \rightarrow \Lambda V$? I suspect this comes from a similar reason to that explained  by Prahar, but I would appreciate a confirmation about this. What Is the Real Name of the #? How do you find the value of $\sum_{r=0}^{44} \tan^2(2r+1)$? Problem:   Find the value of $$\sum_{r=0}^{44} \tan^2(2r+1)$$  Note: The angles here are in degrees.    I don't know how to solve this question because trigonometric simplifications didn't get me anywhere. I think there was a method to solve this question using complex numbers which I no longer remember. Any hint/help will be appreciated. How to correctly do a division using a slash? What's the correct way to do division using a backslash? If I write $a/bc + d$, will that be equal to $(a/(bc))+d$?   Basically, if I place a slash, will I divide by what's directly behind the slash $(b)$, the term that's behind the slash $(bc)$, or everything after the slash $(bc+d)$? I seem to have run into a bit of a challenge on one of our Android/Samsung devices...   On the device, regardless of what's open, when the battery reaches 25% on discharge, the below screen pops up and covers the entire screen, with MUCH difficulty to get rid of. I've tried to trace it to a certain app, but was unsuccessful in doing so.     Can someone perhaps help with some advice on this, on how to get rid of it? Why does no physical energy-momentum tensor exist for the gravitational field? How to change the snapping increment in different units? I would like to know how to change the increments used for snapping transforms.  This would be useful in a number of situations but my problem now is that when I change the units it does not scale the snapping increments to match.  So when holding down ctrl the cube snaps from here:    To here:    I would like it to snap to the 1 foot grid, or possibly any other amount I find useful.  It would also be nice to be able to change the rotation snap as well. Which of the following sets are compact in $\mathbb{M}_n(\mathbb{R})$[NBHM_PhD Screening Test-2013, Topology] Which of the following sets are compact in $\mathbb{M}_n(\mathbb{R})$ (a) The set of all upper triangular matrices all of whose eigenvalues satisfy $|\lambda|≤2 $. (b) The set of all real symmetric matrices all of whose eigenvalues satisfy $|\lambda|≤2 $. (c) The set of all diagonalizable matrices all of whose eigenvalues satisfy $|\lambda|≤2 $.  (a) &amp; (c) are not true. But I am not sure about (b).can anybody help me please. Use /testfor to make a custom command Basically, I'm trying to recreate the Faction plugins /f map in my single-player without mods as my PC doesn't run them very well. (I know it sounds strange.) I am trying to use the /testfor to test if I have written '/f map' in chat, and if I have, it would show me a custom generated map. I don't know how to word this better xD Essentially what I want is for it to test if I have written a command in chat, and if I have, display something. I already know that if groups $G,H,K$ are finitely generated abelian groups, following is true.  If $G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$. I prove this by uniquness of factorization of finitely generated abelian groups.  My questions are   If group $G$ is finite,$G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$? Can you give me a easiest proof and it can be proved by projection function on external direct product? If $G,H,K$ are groups.  If $G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$. This statement is false. What is a counterexample?   *$\times$ is external direct product Replacing the nth instance of a regex match in Javascript I'm trying to write a regex function that will identify and replace a single instance of a match within a string without affecting the other instances.  For example, I have this string:  12||34||56   I want to replace the second set of pipes with ampersands to get this string:  12||34&amp;&amp;56   The regex function needs to be able to handle x amount of pipes and allow me to replace the nth set of pipes, so I could use the same function to make these replacements:  23||45||45||56||67 -&gt; 23&amp;&amp;45||45||56||67  23||34||98||87 -&gt; 23||34||98&amp;&amp;87   I know that I could just split/replace/concat the string at the pipes, and I also know that I can match on /\|\|/ and iterate through the resulting array, but I'm interested to know if it's possible to write a single expression that can do this.  Note that this would be for Javascript, so it's possible to generate a regex at runtime using eval(), but it's not possible to use any Perl-specific regex instructions. How to disable the "highlight changes" feature? Our admins deployed a new version of Notepad++ (version 6.3) on our machines, and this version is highlighting all changes done in a file by default. It does so by setting the background color of every modified line to a bright orange, which I find very distracting.  We  have dozens of plugins installed, but I double checked and we do not have the "Change Markers" plugin (which could be the culprit).  Any idea where I can disable this feature? Changing the highlight color could also be an option. How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.  This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out, that proving the above statement is enough for proving the continuity of $\sin$, but I can't find out how. Any help is appreciated. This is a question in general, not specific to any method or data set. How do we deal with a class imbalance problem in Supervised Machine learning where the number of 0 is around 90% and number of 1 is around 10% in your dataset.How do we optimally train the classifier.  One of the ways which I follow is sampling to make the dataset balanced and then train the classifier and repeat this for multiple samples.   I feel this is random, Is there any framework to approach these kind of problems. Assigning point IDs to respective start and end attributes of polyline?