How can I evaluate $\displaystyle\sum_{n=2}^\infty n(n-1)x^n$?   I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer.
My Prof wrote that   $\sum_{n=2}^{\infty} n(n-1)x^{(n-2)} = \frac{2}{(1-x)^3}$  when $-1&lt; x&lt;1$  I do not understand where the above formula comes from. Can someone please explain the steps or provide a hint?   Thanks!
Is it theoretically possible that every person in this village has a different jar of coins? A village has about $1$ million residents. Suppose each resident has a jar with $100$ coins in it.Two jars are considered to be “equivalent” if they have the same number of 10c, 20c, 50c, $\$$1 and $\$$2 coins in them. How many different (i.e. nonequivalent) jars of coins exist? Is it theoretically possible that every person in this village has a different jar of coins?  I understood that if we have $k_1$ 10 coins in the jar where $k_1$ is between $0$ and $100$ inclusive, then there are $100-k_1$ coins of other denominations in the jar. Likewise, we also have $k_2$ 20c coins in the jar where $k_2$ is between $0$ and $100$ inclusive, so the combined number of 50c, $\$$1, and $\$$2 coins in the jar is $100-k_1-k_2$. And we always have some $k_1, k_2, k_3, k_4, k_5$, let's say, so that the sum of those $k$'s is $100$.  I got stuck since I don't know how to 'compare' the number of coins in jars for different residents. List user's folder access permissions Does anyone know any sources for free European city boundary data? How to mount a virtual hard disk? Is it possible to mount a virtual hard disk (VHD, HDD, VDI, VMDK) on Ubuntu? How can this be done? Ubuntu installer can't find any disk on Dell XPS 13 9350 I am trying to install Ubuntu 14.04 / 15.10 on my Dell XPS 13 9350 (November 2015).  I created a bootable USB disk which boots fine but the installer and the file manager do not detect the hard drive.  What is the problem? How can I solve it? Proving a trig identity: $\frac{\sin(A + B)}{\sin(A - B)}=\frac{\tan A + \tan B}{\tan A - \tan B}$ How do I troubleshoot the "Error, press shutter release button again" message on my Nikon? The LCD on my Nikon D3000 is displaying the following error message:     Error. Press shutter release button again.   I have pressed the shutter release button but the camera still does not work. Soft Thresholding CUDA implementation Does white whole wheat flour need to be refrigerated? I know that regular whole wheat flour needs to be refrigerated, but does white whole wheat flour also need refrigeration? I hate to use up the space in the refrigerator if I don't have to.  I have 5 lb. (2.3 kg), and it would probably take me a month or so to use it all. Disabling submit-button until the user has completed the form How do you think about disabling the submit-button until the user has completed the form or the required fields?   I find similar questions here on ux.se and somebody answered that the user will think that the form is broken or not immediately realize whats going on.  I think hiding is not that good but a disabling (e.g. grayscale, opacity, ...) plus a short text in the button would improve usability because the user realise that he still has to fill out something.  Is there anybody with experience or examples besides the usual error-messages/validation after submitting? How do I install VMware player on a Mac os X? What is "it" in the following sentence: It is clear that Bob likes doughnuts I am very confused. Unless I am mistaken, I know "it" has to be a noun of some sort, but I am unable to figure out what noun "it" is referring to.   What is "it" in the following sentence:      It is clear that Bob likes doughnuts.   Heres another couple of examples:     It is impossible to fly. I have a script that needs to do some stuff based on file creation &amp; modification dates but has to run on Linux &amp; Windows.  What's the best cross-platform way to get file creation &amp; modification date/times in Python? Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<|\mathbb{R}|$, is it true that $X$ is null? E:Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages How can I stop Ubuntu Dock from extending to the edges? By default Ubuntu Dock in Ubuntu 17.10 and later releases always extends to the edges (top-bar-to-bottom if placed on the left or right and left-to-right when placed at the bottom) even when it's not filled with icons. Is there a way to prevent this behaviour (i.e. I want to make the dock-length flexible so that as the number of application icons in the dock increases, it should expand accordingly)?  ➡️➡️➡️  In other words, is there a way make Ubuntu Dock look like a &quot;proper dock&quot; instead of a side/bottom panel with full-height/width? Where can I download Windows 8 legally, from Microsoft? What is the square root of $i^4$? What is the $\sqrt{i^4}$?   $i^4$ = $(i^2)^2$ So is $\sqrt{i^4}$ = $\sqrt{(i^2)^2}$ = $i^2$ = $-1$?  Or is $\sqrt{i^4}$ = $\sqrt{1}$ = $1$?  When I plug it into my TI-89 Titanium, I get $1$.  Edit: I now see why $i^4 \neq -1$. $$\sqrt{x^2} = \left|{x}\right| $$ So$$\sqrt{(i^2)^2} = \left|i^2\right| = \left|-1\right| = 1$$ Why crontab scripts are not working? Show that  $$t(n) = 1 + \sum_{ j=0}^{n-1} t(j)$$  is the same as  $$t(n) = 2^n$$  Initial condition $t(0) = 1$