$$\lim_{x\to 0}\left( \frac{1^x+2^x+3^x+\cdots+ n^x} n \right)^{a/x} $$
This equation is directly given in my book and I am don't know anything about its proof.I tried L'Hospital rule by differentiating the both numerator as well as denominator(division rule), but the result is still coming in indeterminate forms.I am a beginner , and haven't practiced limits that much. This formula is really confusing me.
When did the 5-foot square/hex become the standard unit of measurement in Dungeons & Dragons? Custom ordering of duplicate targets is not necessarily honored on mobile apps I want to prove that if we have a commutative integral domain $D$ with field of fractions $F\neq D$ then $F$ is not finitely generated as a $D$-module. (In this question it may be the case that $1\not\in D$.)    My original plan was to localize at a maximal ideal and then use Nakayama's lemma but as we don't have a $1$ our localization may not be local (in fact we need not have maximal ideals).  So I'm a bit lost as to where to go?  Thanks for any help. Proving with induction $(1-x)^n<\frac 1 {1+nx}$ Prove using induction that $\forall n\in\mathbb N, \forall x\in \mathbb R: 0&lt;x&lt;1: (1-x)^n&lt;\frac 1 {1+nx}$   My attempt:   Base: for $n=1: 1-x&lt;\frac 1 {1+x}\iff 1-x^2&lt;1$, true since $0&lt;x&lt;1$.  Suppose the statement is true for $n$, prove for $n+1$:  $(1-x)^{n+1}=(1-x)(1-x)^{n}\overset{i.h}&lt;\frac{(1-x)}{1+nx}$  Now I got stuck, maybe another induction to show that $1+nx+x&lt;1+nx$? Is there another way?  Moreover, I was told it's wrong to begin with $(1-x)^{n+1}$ and reach to $\frac 1 {1+(n+1)x}$ but why? Is it assuming what I need to prove? I recently upgraded a c# windows service to run as a 64 bit .net process. Normally, this would be trivial, but the system makes use of a 32-bit DLL written in C++. It is not an option to convert this DLL to 64 bit, so I wrapped the DLL in a separate 32 bit .net process and exposed a .net interface via remoting.  This is quite a reliable solution, but I would prefer to run the system as a single process. Is there any way I can load my 32 bit DLL into a 64 bit process and access it directly (perhaps through some sort of thunking layer)? How do I programmatically log in a user with a POST request? Approximating $\pi$ with least digits Do you a digit efficient way to approximate $\pi$? I mean representing many digits of $\pi$ using only a few numeric digits and some sort of equation. Maybe mathematical operations also count as penalty.  For example the well known $\frac{355}{113}$ is an approximation, but it gives only 7 correct digits by using 6 digits (113355) in the approximation itself. Can you make a better digit ratio?  EDIT: to clarify the "game" let's assume that each mathematical operation (+, sqrt, power, ...) also counts as one digit. Otherwise one could of course make artifical infinitely nested structures of operations only. And preferably let's stick to basic arithmetics and powers/roots only.  EDIT: true. logarithm of imaginary numbers provides an easy way. let's not use complex numbers since that's what I had in mind. something you can present to non-mathematicians :) The Monty Hall problem I was watching the movie 21 yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the wrong answer) which revolves around theoretical probability.  The question goes a little something like this (I'm paraphrasing, but the numbers are all exact):  You're on a game show, and you're given three doors. Behind one of the doors is a brand new car, behind the other two are donkeys. With each door you have a $1/3$ chance of winning. Which door would you pick?  The character picks A, as the odds are all equally in his favor.  The teacher then opens door C, revealing a donkey to be behind there, and asks him if he would like to change his choice. At this point he also explains that most people change their choices out of fear; paranoia; emotion and such.  The character does change his answer to B, but because (according to the movie), the odds are now in favor of door B with a $1/3$ chance of winning if door A is picked and $2/3$ if door B is picked.  What I don't understand is how removing the final door increases the odds of winning if door B is picked only. Surely the split should be 50/50 now, as removal of the final door tells you nothing about the first two?  I assume that I'm wrong; as I'd really like to think that they wouldn't make a movie that's so mathematically incorrect, but I just can't seem to understand why this is the case.  So, if anyone could tell me whether I'm right; or if not explain why, I would be extremely grateful. Use fixed keyboard for a given app I would like to tell my Android that when I'm in a given app it should switch to a given keyboard. My use case is that I want to use Hacker keyboard with Connectbot. Much as I loathe Microsquish's Word, it does allow you to insert into a header or footer the automated text "Page X of Y."   When using InDesign, it's easy to set up the "Page X" part on a master page. But is there any placeholder character for the Y? That is, it will count the number of pages in the document, either in an absolute or a relative-to-the-section fashion? Every now and again I narrowly miss sending out something with a footer which reads "Page 4 of 3," and I would love to know if there's some way around that. Can't use method return value in write context Can I achieve a torchlight effect (lighter area around a light source) in a 2D game? I am thinking of writing myself a simple 2D game. It will not shine with perfect graphics or gameplay at first, but I'd consider it my first step in PC game development. So, imagine such simple sprite-based 2D game (like Heroes IV or Startcraft BroodWar).  I want the gameplay to support day/night with the according lighting changes and at the same time it will be madness to have to create sprites for every lighting nuance. So, I decided adding a semi-transparent layer on top of other objects will be enough.   The issue with this solution is if I have a light source object in the game (like the hero wearing a torch, or a burning building), there must be a lighter area around it, right? Since I am putting my semi-transparent layer over everything, how would you suggest to achive the torchligt visual effect I want? Maybe redraw that layer adding 'gaps' or differently colored areas based on the lighting effect? Suppose f is a function such that $f'(x) = \frac{1}{x}$ for all $x &gt; 0$. Prove that if $f(1) = 0, then f(xy) = f(x) + f(y)$ for all $x,y &gt; 0$  As stated above, how do i go about doing this question?   Hi all, i would like to apologize if the question is missing context or other details.   This question was part of my calculus tutorial. I was on the topic of application of differentiation when this question was given to me.  Thus far i have learnt about intermediate value theorem, mean value theorem, rolle's theroem, Fermat's thereom.  As of this point in my learning, i have not touched upon integration. The only reason why i accepted the answer with integration in it was because i understood it.  Thus i believe i am suppose to use one the few theorems that i've stated to solve this question. But i have no idea where to start.  At first i differentiated the equation $f(xy)=f(x)+f(y)$ with respect to x  i believed the answer to be $f'(xy)\frac{dy}{dx}=f'(x)+f'(y)\frac{dy}{dx}$  which would be $\frac{1}{xy}\frac{1}{x}=\frac{1}{x}+\frac{1}{y}\frac{1}{x}$  At this point a few questions came to my mind  1) am i suppose to differentiate the eqn in the first place 2) what has $f(1)=0$ got to do with anything 3) what theorem am i suppose to use to solve this question  This was my thought process throughout the question and i hope someone would point out my mistakes. Thank you I have just created a Minecraft 1.13.1 server and was wondering if there was any way to give everyone the teleport command even if they are not op. If not, is there a way that I could make them op but only have access to the teleport command? What's the right way to hard boil eggs? It is well known that compactness implies pseudocompactness; this follows from . I know that the converse does not hold, but what is a counterexample?  (A  is a topological space $S = \langle X,{\mathfrak I}\rangle$ such that every continuous function $f:S\to\Bbb R$ has bounded range.) Method for batch layout of images into PDF sheets I have 100,000 identically-sized, but unique, PNG/JPG images which I want to automatically layout onto A4 sheets which can then be exported as print-ready PDFs.  So assuming I can fit 8 of these images per page in a 4x2 grid, I'm looking to input my 100,000 images and output 12,500 PDFs.  I'm a programmer so I could probably write a script to do this myself (e.g. PHP/Python), but I was wondering whether there are any tools/scripts/software that already exists that can do this automatically. Plot x=3 in PGFPlots Dimensions with Rank-Nullity question $f:V \rightarrow V$, $\dim(V) = n$  a) If $n$ is odd, prove that there does not exist a linear transformation with $\operatorname{im}(f) = \ker(f)$  b) Is it false to affirm the previous statement if $n$ is even?  My answer: $$ i = \dim(\operatorname{im}(f))\\ k = \dim(\ker(f)) \begin{cases} n=2i                           &amp; \text {if $i=k$} \\ n=i+k   &amp; \text{if $n\neq k$} \\ \end{cases}  $$  b) Yes.  Is that a valid proof? >Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$. Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$.   I saw some proof of $\frac d {dx} x^n=nx^{n-1}$ using binomial theorem, which is only available for $n \in\mathbb N$. Do anyone have the proof of $\frac d {dx} x^n=nx^{n-1}$ for all real $n$? Thank you.