I've calculated $\lim_{n\to\infty}\dfrac{1^p+2^p+\cdots+n^p}{n^{p+1}}=\dfrac1{p+1}$ where $p\in\mathbb{N}$ fixed. I feel it should help me get this one $\lim_{n\to\infty}\left(\dfrac{1^p+2^p+\cdots+n^p}{n^{p}}-\dfrac{n}{p+1}\right)$, but I'm not sure how. Any hints?
Evaluate  $$\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$$
What triggers the benefit of the bard's "Song of Rest" feature? In D&amp;D 5e, the benefit of the Bard's "Song of Rest" feature triggers "if you or any friendly creatures who can hear your performance regain hit points at the end of the the short rest."  Does Song of Rest's benefit apply just when spending hit dice to regain hit points? Or does it apply to any other way that a player might regain hit points at the end of a short rest (e.g. by casting a spell, using a Second Wind, eating a Goodberry, etc.)? Is a constant condition more costly than switching shaders? In general, branching in shaders is not a good idea. But now I have a shader with a condition that is constant with respect to the entire draw call. So the branch that is executed is always the same for one draw call.  Is such kind of branching still more costly than having multiple shaders without these branches and switch between them? Does there exist a magical shell piping which would allow easily to grep through bunch of .gz log files without needing to extract them somewhere?  .gz files are Apache logs, result of log rotation. I'd like to quickly check how often certain URIs are accessed in the past. How to resize partitions? Previously, I have installed Windows 7 on my 320 GB laptop with three partitions 173, 84 and 63 GB each. The 63 GB partition was where the Windows was installed. The rest were for file containers.  Now I changed my OS to Ubuntu 12.04 LTS. I installed Ubuntu by replacing the entire Windows 7 on the 63 GB partition. The rest of the partitions remain as an NTFS Windows partition and I can still access them both (the 173 and 84 GB partitions).  Now I want to change the two partitions of Windows into an Ubuntu format partitions plus most importantly, I want to extend the 63 GB partition to more than a 100 GB because at the moment I am running out of disk space. 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However, I found  referring to operators under  that allowed me to add sensors and controllers from python.  bpy.ops.logic.sensor_add(type='ALWAYS', object=my_obj.name) bpy.ops.logic.controller_add(type='LOGIC_AND', object=my_obj.name)   However I am not able to find any way to link them from python. I tried following (but did not work):  new_controller = my_obj.controllers[-1] new_sensor = my_obj.sensors[-1]   It throws AttributeError saying that object has no attribute 'controllers' :-/ On a series involving harmonic numbers Finding $$\sum^{\infty}_{n=1}\bigg[\bigg(\frac{(-1)^{n+1}}{n+1}\bigg)\bigg(\sum^{n}_{r=1}\frac{1}{r}\bigg)\bigg]$$   Try: Let $\displaystyle H_{n} =\sum^{n}_{r=1}\frac{1}{r},$ then series $\displaystyle \sum^{\infty}_{n=1}\frac{(-1)^{n+1}}{n+1}\cdot H_{n}=\frac{H_{1}}{2}-\frac{H_{2}}{3}+\frac{H_{3}}{4}\cdots\cdots$  could some help me to solve it, thanks is incompatible with the enumerate and itemize environments. Is there a typographical reason not to wrap a list around a figure or is it a technical limitation of LaTeX? If it is a technical limitation, is there any work (new packages or changes to LaTeX) currently being done to eliminate the limitation in the future?  Edit: To respond to some of the comments: I realize that there are "gymnastics" which can make this work. I do not have an example of the problem. Whenever I run into a problem with wrapfig because it is near an enumerate or itemize environment, I search for an easy solution, find none, and move the figure elsewhere. In the vast majority of cases, the cost of learning the complex solutions isn't worth the benefit of keeping the figure where I want it. My Minecraft app now only has the option of "play" and there is no "play on realms" to be found!  I downloaded - and paid for - the multiplayer app, but there doesn't seem to be a way to get to any realms on that app either. Show that $\deg f$ is even when $n$ is odd. A map $f:S^n\rightarrow S^n$ satisfying $f(x)=f(-x)$ for all $x\in S^n$ is said to be an even map. Show that if $f:S^n\rightarrow S^n$ is an even map, then $\deg f=0$ when $n$ is even and $\deg f$ is even when $n$ is odd. Moreover show that when $n$ is odd, there exist even maps of any given even degree.   My Try:  So $f=(-I)\circ f$. Then $\deg f=\deg (-I)\times \deg f$. If $n$ is even $\deg (-I)=-1$. Hence, $2\deg f=0$, so $\deg f=0$. But how do I show the results when $n$ is odd. No clue at all. This problem is in Hatchers book (A very complicated text). There is a hint but I do not understand it. Can somebody help me to proceed? ¿Can you help me with axiom of regularity? Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$ How to prove the following product? $$\frac{\sin(x)}{x}= \left(1+\frac{x}{\pi}\right) \left(1-\frac{x}{\pi}\right) \left(1+\frac{x}{2\pi}\right) \left(1-\frac{x}{2\pi}\right) \left(1+\frac{x}{3\pi}\right) \left(1-\frac{x}{3\pi}\right)\cdots$$ Multiplication principle and permutation Definition of Multiplication principle in Principles and Techniques in combinatorics by Chuan-Chong, Khee-Meng is given as:      Let $$\prod_{i=1}^rA_i=A_1\times\dots\times A_r=\{(a_1,\dots,a_r) | a_i\in A_i, i=1,\dots,r\}$$ denote the cartesian product of the finite sets $A_1,\dots,A_r.$ Then $$\left\vert \prod_{i=1}^r A_i \right\vert=|A_1|\times\dots\times|A_r|.$$   I want to know how this principle is applies in defining $P_r^n=n(n-1)\cdots2\cdot1$, ie, what are sets $A_1,\dots,A_r$ here? Understanding the difference between two theorems related to $|G\,:\,K|=|H\,:\,K|\cdot|G\,:\,H|$. Prove that $B[x] \cap B[x^{-1}]$ is integral over $B$ Nikkor f/1.8G 35mm or 50mm? I'm new to photography. I got a Nikon D3100 with the 18-55 kit lens.  I was looking to buy a f/1.8 prime lens, and I found two options:   AF-S DX NIKKOR 35mm f/1.8G   AF-S DX NIKKOR 50mm f/1.8G   How should I go for choosing one or the other? White's test for heteroskedasticity in R I am trying to estimate heteroskedasticity in R. I had Eviews available in my college's lab but not at home. I have been trying to use "het.test" package and whites.htest but the value that I get is different from what I get in Eviews. According to the author of the package, it is meant to do the same test as is done in Eviews.   My model looks like A~B+C. Kindly guide me what I am doing wrong.   I 1st run ols for equation A~B+C, after getting results, I clicked View&gt;Residual diagnostics&gt;Heteroscadicity tests&gt;White's (with Include white cross terms) and then Press ok. The result i got was Prob. F(5,27) = 0.2948.   For R, i tried code at this link [this link][1], here is the edited code that i used.  library(vars)  library(het.test) dataset &lt;- data.frame(A,B,C))  model1 &lt;- VAR(dataset, p = 1)  whites.htest(model1)   This gave me value of 0.63.  I tried searching this forum but none question seemed related. I am not a programmer not statistician so please pardon my ignorance. What is appropriate to use here, who or which?       There are around 50 companies who/which deliver scanning services to private and business consumers.