Prove that $\lim_{n\to\infty} \frac{a^{1/n} - 1}{1/n} = \ln a$
Trying to show that $\ln(x) = \lim_{n\to\infty} n(x^{1/n} -1)$
I have a graphic on the clipboard on OS X.  What's the quickest way to get it onto disk as a png or jpg? If $ \langle G, \star \rangle $ is an abelian group, then for all $a, b \in G$, show that $(a \star b)^{n} = a^{n} \star b^{n}$. If $ \langle G, \star \rangle $ is an abelian group, then for all $a, b \in G$, show that $(a \star b)^{n} = a^{n} \star b^{n}$.  I am stuck at the first step, unable to figure out how to start. I am just beginning on group theory, and know that an abelian group obeys commutativity, associativity, has an identity element, and each element is invertible. I am unable to figure out how to apply this for the proof. Any hints on how I should get started? I understand that when I press and release the power button on the top of my iPhone or iPad, it stays on but turns off the screen.  Whereas if I press and hold the power button, a "slide to turn off" control appears on the screen.  Sliding this, as I was told, does turn the unit off, but it does so by suspending the RAM to disk (flash memory, actually.) So, when turning on the iPhone or iPad, it isn't so much rebooting the OS as it is resuming it from flash storage.  How to force the OS to completely reboot on these devices? Why are my objects in all layers? Repeating something with (1/n)th chance of success n times Is there anything that can be said about how many attempts it will take to correctly guess a random number out of 1000 numbers? If the number wouldn't change the probability would just increase every guess, giving in the end a probability of 1.  If the number changes every guess tough, you just have a 1/1000th chance everytime. Can it be calculated how many attempts it would probably take, on average?  (I'm sorry, i'm not very skilled in probability maths) Show $100+t$ is reducible in $\mathbb{Z}[[t]]$ $\mathbb{Z}[[t]]$ is the ring of formal power series with integer coefficients, and the problem asks to show that $100+t$ is reducible. This means I need to find two power series, $\sum_{i=0}^\infty a_i t^i$ and $\sum_{j=0}^\infty b_j t^j$ with $a_i,b_j\in\mathbb{Z}$, that when multiplied give $100+t$. Since the integers are an integral domain, I know that neither of these power series may be finite, but I'm having a really hard time figuring out how to come up with two series that will telescope properly. Any suggestions? What if I run a virus/trojan Windows EXE on Ubuntu with Wine? Uniform convergence of derivatives, Tao 14.2.7. Can I use a HDD on my PS3 if I have already used another on the same console? I'm thinking of changing the Hard Disk of my PS3 (Slim 120GB), but I am unsure of what will happen to the data on the one I previously had if I do this:   PS3 has some data in the HDD. I turn off PS3, change HDD, install OS on new HDD. I download some data on new HDD from PS Plus.   Let's imagine I forgot to back something up...   I turn PS3 off, change HDD back.   Can I still use the old hard drive without needing to format it again? In Windows, you can run android application using BlueStack.   I was wondering, is there any way these apps can be run on Ubuntu 12.04? Solve $x^2+5x+6 \equiv 0 \pmod{\!11\cdot 17}$ Solve $x^2+5x+6 \equiv 187 \mod 187$ Solution $$x^2+5x+6 \equiv 187 \mod 187$$ $$ (x+\frac{5}{2})^2 \equiv \frac{1}{4}$$ $$ 4(x+\frac{5}{2})^2 \equiv 1$$ $$ y:= x+\frac{5}{2} $$ $$ 4y^2 \equiv 1 \mod 11 \wedge 4y^2 \equiv 1 \mod 17  $$ $$ ( 2y \equiv 1 \mod 11 \vee 2y \equiv 10 \mod 11 ) \wedge ( 2y \equiv 1 \mod 17  \vee 2y \equiv 13 \mod 17)  $$ $$ ( y \equiv 6 \mod 11 \vee y \equiv 5 \mod 11 ) \wedge ( y \equiv 9 \mod 17  \vee y \equiv 15 \mod 17)  $$ Combining that from CRT I got: $$ y \in \left\{49, 60,83,94 \right\} $$ and for example: $$ x+\frac{5}{2}  \equiv 94 \mod 187$$ $$ 2x  \equiv 183 \mod 187$$ some calculus and get... $$x \equiv 185 $$ And the same thing for each other case. Question Is there any faster (or smarter) way to solve equations like that? Prove $\int_{a}^{b}{f(x)dx}>0$. Let $f\ge0$ be integrable on $[a,b]$. Let $f$ be continuous on $x_0\in (a,b)$ and let $f(x_0)&gt;0$.   Prove $$\int_{a}^{b}{f(x)dx}&gt;0$$  It is easy to see it, graphically, because $$f\ge 0 \Rightarrow\int_{a}^{b}{f(x)dx}\ge0$$  and if $f&gt;0$ somewhere between $a$ and $b$, then the zone of $F$ between those point is "real" (positive) and $\ne 0$.  My problem is, how do I show it formally?  I would appreciate your help. Divisor function asymptotics Define $\tau_{r}(n) = \sum_{d_1...d_r = n}1$.   One exercise in a book on sieve theory asked for an elementary proof by induction of the fact that   $$\sum_{n\le x}\tau_r(n) = \frac{1}{(r - 1)!}x(\ln x)^{r - 1} + O\left(x(\ln x)^{r - 2}\right)$$  The base case $r = 2$ is easy with reversing the order of summation.  The only other progress that I made was the fact that $\sum_{n\le x}\tau_{r}(n) = \sum_{d\le x}\left\lfloor\frac{x}{d}\right\rfloor\tau_{r- 1}(d)$, but I don't know how to proceed. Why are integer literals with leading zeroes interpreted strangely? This prints 83  System.out.println(0123)   However this prints 123  System.out.println(123)   Why does it work that way? What are tag synonyms and merged tags? How do they work? What are tag synonyms? How can tag synonyms be distinguished from "real" tags? How are new tag synonyms created? Who can create tag synonyms? How can I delete/reverse/undo bad tag synonyms? What are merged tags? Can you grapple/shove with the Hunter Ranger's Whirlwind Attack? The  states:     You can use your action to make a melee attack against any number of creatures within 5 feet of you, with a separate attack roll for each target.   The rules on "" state (the rules in shoving are effectively identical):     When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.   I couldn't find any other features spells or abilities that allow a creature to make a flat "melee attack". If they had said "melee weapon attack" instead this would have meant the same thing but specifically have excluded grapple/shove as an option.  I am unsure whether the rules on grappling/shoving mean that the only time you can grapple/shove are when you take the Attack action, which you are not doing when using the Whirlwind Attack feature.   I would like answers to assume that the following ruling made in the  is correct:     Whirlwind Attack is unusual, in that it’s a single attack with multiple attack rolls... What's the difference between  var A = function () {     this.x = function () {         //do something     }; };   and  var A = function () { }; A.prototype.x = function () {     //do something }; An example for a homomorphism that is not an automorphism Convergence of $\int_0^\infty $sin$ (x^p) dx$ Show that $\tan{(\pi/7)} \tan{(2\pi/7)}\tan{(3\pi/7)}=\sqrt{7}$