The question is as follows:  The power series $$\sum _{ k=0 }^{ \infty  }{ k{ z }^{ k } } $$ represents an analytic function $f$ in its disk of convergence, the disk of radius $R=1$, centered at the origin. Find $f$.  The answer is:  $$f\left( z \right) =\frac { z }{ { (1-z) }^{ 2 } } $$    I proceeded by a hand-crafted method, ie. I wrote down the beginning of $\sum _{ k=0 }^{ \infty  }{ k{ z }^{ k } } $, and just below the $\sum _{ k=0 }^{ \infty  }{ { z }^{ k } } $ and take the sum of them term by term. It yielded into an expression already known:  $$f(z) (= \sum _{ k=0 }^{ \infty  }{ k{ z }^{ k } }) = \sum _{ k=0 }^{ \infty  }{ (k+1){ z }^{ k } } - \sum _{ k=0 }^{ \infty  }{ { z }^{ k } } $$  As,  $$ \sum _{ k=0 }^{ \infty  }{ { z }^{ k } } =\frac { 1 }{ 1-z } \quad (1)$$ and $$ \sum _{ k=0 }^{ \infty  }{ (k+1){ z }^{ k } } =\frac { 1 }{ { (1-z) }^{ 2 } } \quad (2)$$  I reached the solution.  But I feel like cheating, because the conclusion about $(2)$ was evaluated in the course; also my instructor used another method for this question: She differentiated the power series and manipulated analyticity. I also tried to integrate first, then differentiate by regrouping terms, but neither could I achieve nor could I understand the professor's method.  I'd much liked you help me to find alternative solution -as my professor did- to this?
Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$ how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2. $$   Thank you
Is there a rising trend in Number Series I am trying to build a stock screening utility.   What  I am trying is to find if there is a rising trend in a time series of profit margins of a company. I know there can be dips in some years but I want to find if mathematically that overall there is a rising trend as well as towards the recent years there is higher margins as compared to previous years.   Is there any way we can find it mathematically?          Profit Return Year,Company A,Company B,Company C 2005,38.97,20.44,22.01 2006,22.4,20.59,20.63 2007,22.4,20.59,20.63 2008,21.56,22.01,23.87 2009,25,20,23.52 2010,25,22.38,18 2011,30,24.13,24.85 2012,32,25.43,27.47 2013,30,20,22 2014,35,27,20 I've installed Process Lasso and in All processes tab, set all affinities to 1-7, meaning that all the processes can be start on all my CPUs except the first one. Yet, there is a huge load on my first CPU.   I personally have no clue what's happening here and really appreciate any help.  My system is:   Intel core i7 - 4720 HQ  2.6 GHz upto 3.6 Windows 10 (64 bit) NVidia GTX 950 M   This screenshot, shows the software's interface. Those 8 green bars on upper right show my CPU usage. Also the column called Rules, indicates the affinities on the processes.       PS: These are screenshots that show the more familiar Task Manager and Resource Monitor. I've sorted all with respect to their CPU usage. Note that the overall CPU usage is just fine (less than 30% most of the time), but in Resource Monitor, same as the Process Lasso, it's obvious that that most of the load is on CPU0 and the others are almost free.  Also I found out that my total CPU usage, is actually more than all my processes altogether! The orange line in Resource Monitor picture, shows the amount of CPU usage of all my processes. (I did the same in safe mode. Orange line, exactly coincides the total usage in safe mode. So I'm suspecting that something is happening in my normal boot.) Who is bound to the Prime Directive? In the galaxy there are a fair number of pre-warp societies. A Federation ship is not to contact them. But what about other, non-Federation species?   If for example a Ferengi ship passes by a pre-warp planet, would they be allowed to contact it? What would a Federation ship observing this contact have to do? Are they obliged to intervene? Are they allowed to? Or were they merely allowed to "strongly discourage" the attempt of the Ferengi?  If it is legal to use force to stop the Ferengi, than in essence the Ferengi were bound to the Prime Directive too. And if so: does this go for all Federation territory? Is this confined to the Federation territory? If Janeway observes such an incident, what are her rights to intervene? Or does she have the duty to do so? Let $f:X\to Y$ be a function and $E\subseteq X$ and $F\subseteq X$.   Show that in general $f(E − F)\nsubseteq f(E) − f(F)$.   I have no idea about how to prove this; and could anyone please explain the basic theory of functions(by relating to this question. Thank you. What’s the rule for using “who” and “whom” correctly? I can never figure out whether I should use who and whom. Most people use who for both colloquially, but some people say this is not correct.  What’s the rule for using who and whom correctly? Where does gravity get its energy from? Smooth shaded model has odd shading I'm currently working on a model of an X-wing that I'm intending for use in a game and running into a bit of trouble with the shading of the mesh. The upper nose is rounded off where the sensor cone attaches, but at the point it reaches the canopy frame it flattens out and meets at a corner.  Everything is fine and dandy when the sides are modeled:    but once I close it off and add the top of the nose, the shading goes to hell:    As you can see, the smoothing gets confused at the lip of the cockpit.  I've tried a couple different means of getting around this, none of which looks right while maintaining the correct shape. The transition from the rounded to flat is a fairly important part of the ship's character, so I don't want to discard the contours, either.  Any tips or suggestions on modeling this area?  Addendum:  If I eliminate that second angle where the canopy frame meets the fuselage as marked in red:    this also fixes the smoothing. However this is not a solution, as the angle is a part of the design. It DOES give me another thought on fixing, but I won't be able to test it until I get home. If I have a 18-55mm lens, is there a point in buying a 35mm prime lens? Why so many files in /usr/share/app-install/desktop? For 14.04, I'm trying out Lubuntu LTS.  If I run ls -l /usr/share/app-install/desktop, I get a list of 3085 .desktop files. All except one are dated Apr 18. The exception is:  -rw-r--r-- 1 root root  6328 Mar 28  2013 applications.menu   From the dates, it's clear that all these are installed by default and not by me.   Doesn't each GUI-based program bring in its own .desktop file at the time it's installed (mostly in /usr/share/applications)? Then why are these thousands of other .desktop files needed? From their names, most of them haven't anything to do with Lubuntu. Their names are different from regular .desktop files (such as those in /usr/share/applications): many, not all, are like name:name.desktop whereas the regular ones are just name.desktop. The ones I looked at all have a line with X-AppInstall-Popcon=nnn where nnn is a string of digits. I know that Popcon "gathers statistics determining which packages are the most popular with Ubuntu users". But why is that part of these .desktop files? Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.  So $(a^2+b^2)(c^2+d^2) = a^2c^2+a^2d^2+b^2c^2+b^2d^2$  and $(ac+bd)^2 = a^2c^2+2acbd+b^2d^2$  So the problem is reduced to proving that $a^2d^2+b^2c^2\ge2acbd$ but I am not sure how to show that Catching exceptions from a constructor's initializer list Confirm and agree to negative questions Farm Solution Vs. SandBox Can I use an Android phone as webcam for an Ubuntu device? I've got a utility that outputs a list of files required by a game.  How can I run that utility within a C program and grab its output so I can act on it within the same program? UPDATE: Good call on the lack of information.  The utility spits out a series of strings, and this is supposed to be portable across Mac/Windows/Linux.  Please note, I'm looking for a programmatic way to execute the utility and retain its output (which goes to stdout). Can anyone tell me how Google decides which image to use when it shows an image with the search results (as is the case with mobile search, sometimes)?  If you look at these mobile search results …    … and take the AliExpress.com result as an example, by inspecting  you can see that  used by Google seems to have no special declaration, etc. It is just the first JPG that appears in that page's HTML - so I was thinking maybe that's why it's the image that is used in the Google search results.  However, when you look at these mobile search results …    … and consider the Backpack Billboards result,  reveals that  being used is actually the seventh image declared in the HTML.  So can anyone shed any light on how Google determines which image to be displayed? I would like to know so I can then have control over which image is displayed for my site.  Update / Solution  Thanks to the answers, below, this is the code I have now added to my head HTML in order to get the thumbnail shown in Google when searching for backpack billboards:  &lt;script type="application/ld+json"&gt; {   "@context": "http://schema.org/",   "@type": "Product",   "name": "Regular Backpack Billboard",   "image": [     "https://backpackbillboards.com/wp-content/uploads/2017/12/backpack-billboards-1x1.jpg",     "https://backpackbillboards.com/wp-content/uploads/2017/12/backpack-billboards-4x3.jpg",     "https://backpackbillboards.com/wp-content/uploads/2017/12/backpack-billboards-16x9.jpg"    ],   "description": "Backpack Billboards feature a rechargeable battery which powers the internal LED lights providing a bright illuminated display for up to 8 hours.",   "sku": "BPBB8",   "brand": {     "@type": "Thing",     "name": "Backpack Billboards"   },   "offers": {     "@type": "Offer",     "priceCurrency": "USD",     "price": "299.00",     "priceValidUntil": "2020-11-11",     "itemCondition": "http://schema.org/NewCondition",     "availability": "http://schema.org/InStock",     "seller": {       "@type": "Organization",       "name": "Backpack Billboards"     }   } } &lt;/script&gt; I use  to store all articles I need and bibtex4word add-on (in MS Word) to maintain the reference list. So, I would like to know how to properly fill the "author" field in Jabref with multiple authors to appear them correct in the reference list.  For example, I've got the article with several authors:      E. Orti, J.L. Bredas, C. Clarisse   And I would like them to appear in this manner. But instead I get this:     C. C. E. Orti, J. L. Bredas Existence of elements of order $p^k$ in a finite group By Cauchy's theorem, if a prime number $p$ divides the order of a group $G$, then there is an element $g$ in $G$ whose order is $p$. In addition, if $|G|$ admits a prime decomposition with a factor $p^k$, then Sylow's theorem tells us that a subgroup of order $p^k$ must exist.   My question is the following. If $|G|$ admits a prime decomposition with a factor $p^k$, does this imply that an element of order $p^k$ must exist in $G$?   Update: As explained in Mindlack's answer (among others), the implication does not hold and simple counterexamples exist. In the comments, Mindlack discusses special cases where such an element exists. How to read negative radians in the interval? Say a function's domain is $\{-\pi/2, \pi/2\}$. How should I interpret this interval?  It starts from where? To where? In what direction?  And if it starts from $3\pi/2$, would the next one be $-5\pi/3$.(because it starts from negative, $-\pi/2$). Prevent malicious bots from posting spam