Any known conclusions for $\sum_{k=0}^\infty k^n x^k$ where $n$ is a nonnegative integer and $0&lt;x&lt;1$?   If $n=0$, the sum is simply $\frac{1}{1-x}$. If $n=1$, the sum can be computed as shown in  For general $n$, it seems the method by taking derivative is extremely complicated though I believe the sum would have a finite value because $x^k$ decreases faster than $k^n$ increases.
How can we evaluate this series  $$\sum_{n=1}^\infty \frac{n^a}{b^n}?$$  Here $a$ and $b$ are positive integers. If $b=1$ then series will be diverging, in other cases, it will be converging, but how to find this sum?  Isn't there a simple solution which doesn't involve Stirling numbers?
How can I figure out what's slowly eating my HD space? Every few days I get notices on my MacBook that it's running or run out of hard drive space. Curiously, restarting the computer will enable me to recover gigabytes of space (this past time, it was able to recover about 2.2GB). However, I can't identify anything in my personal activity that consumed that space.  It's possible that it's a rogue iTunes podcast or a huge software update that my Mac is automatically downloading - would either of these reclaim the space upon a restart?  One possibility that I can think of is that FileVault has some sort of disk leak, allocating but not freeing files. Does this make sense? Is there a tool that I can run to determine where this space is going? Assuming it is FileVault, should I try to disable it? What's the best way to turn of FileVault on a nearly full computer? Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$ I'm not sure how to go about this. Does it involve using MVT?  I got as far as saying $e = \frac{e^x - e^y}{x-y}$. CASESAFEID() apex equivalent Is there any function in Apex which could emulate functionality of the  formula function? I'm trying to get 18-char Id variant from the standard one. Can a Half-Elf have Drow Ancestry? Let us say we have the following Makefile  include config.mk  # Generate summary table. results.txt : $(ZIPF_SRC) isles.dat abyss.dat last.dat     $(ZIPF_EXE) *.dat &gt; $@  # Count words. .PHONY : dats dats : isles.dat abyss.dat last.dat  %.dat : books/%.txt $(COUNT_SRC)     $(COUNT_EXE) $&lt; $*.dat  .PHONY : clean clean :     rm -f *.dat     rm -f results.txt   where all the variables are defined in config.mk  LANGUAGE=python COUNT_SRC=countwords.py COUNT_EXE=$(LANGUAGE) $(COUNT_SRC) ZIPF_SRC=testzipf.py ZIPF_EXE=$(LANGUAGE) $(ZIPF_SRC)   Is there any tool available that can expand all the include lines within the makefile so as to produce a bigMakefile? If such a tool does not currently exist, can a python/perl/awk/bash script be cooked up to achieve this goal? 1.  Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an explanation that I need some help with: that the de Broglie wavelength and the wavelength of an elastic wave do not show similar properties under a Galilean transformation. He basically says that both are equivalent under a gauge transform and also, separately by Lorentz transforms. This, accompanied with the observation that $\psi$ is not observable, so there is no "reason for it being real". Can someone give me an intuitive prelude by what is a gauge transform and why does it give the same result as a Lorentz tranformation in a non-relativistic setting? And eventually how in this "grand scheme" the complex nature of the wave function becomes evident.. in a way that a dummy like me can understand.  2.   A wavefunction can be thought of as a scalar field (has a scalar value in every point ($r,t$) given by $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{C}$ and also as a ray in Hilbert space (a vector). How are these two perspectives the same (this is possibly something elementary that I am missing out, or getting confused by definitions and terminology, if that is the case I am desperate for help ;)   3.   One way I have thought about the above question is that the wave function can be equivalently written in $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{R}^2 $ i.e, Since a wave function is complex, the Schroedinger equation could in principle be written equivalently as coupled differential equations in two real functions which staisfy the Cauchy-Riemann conditions. ie, if $$\psi(x,t) = u(x,t) + i v(x,t)$$ and $u_x=v_t$ ; $u_t = -v_x$ and we get $$\hbar \partial_t u  = -\frac{\hbar^2}{2m} \partial_x^2v + V v$$ $$\hbar \partial_t v  = \frac{\hbar^2}{2m} \partial_x^2u - V u$$ (..in 1-D) If this is correct what are the interpretations of the $u,v$.. and why isn't it useful. (I am assuming that physical problems always have an analytic $\psi(r,t)$). I am looking for easy(!) counterexamples that the product of two quotient maps is not necessarily a quotient map and that the quotient space of a Hausdorff space is not necessarily Hausdorff. Why Multiplication of zero by any number is zero? If I multiply $a$ by $0$, it means $a$ is added $0$ times, but I am not able to visualize $0$ times! Let $M$ be the space of all $m\times n$ matrices. And $C=\{X\in M|\operatorname{rank}(X)\leq k\}$ where  $k\leq \min\{m,n\}$. Check whether the set $C$  is:   Closed Connected Compact Open   What are some other good properties of the set $C$,for example is it a manifold?  Clearly the set $C$ is closed  if someone is interested  a good proof can be found , hence $C$ is not open. Also as $C$ is unbounded therefore not compact. How to check whether the set $C$ is connected or not? Formatting a bootable USB that thinks it's a CD-ROM After upgrading to Windows 10, I remember that I had an old bootable recovery usb drive from the manufacturer that I no longer needed. I want to format it but it's showing up as a CD-ROM and format options are not available.  I also tried formatting it in a live linux mint environment, but linux cannot even mount the cd-rom.  How can I format this recovery usb that is showing up as a cd-rom? New SSD problem I've just installed a new SSD replacing an HDD in my old (late 2008) iMac. When I switched it on all I got on screen was a question mark.   On the advice of my son I tried cmdR during switch on with no result.   I then tried alt which at least got me a cursor on the screen, but nothing else. Any suggestions? I've got everything backed up on an external Hard Drive, by the way. How do I add the kernel PPA? Skate shoes vs mtb shoes? It's an old argue on forums, still cant get it. What is the (real, that worth the money and time for ordering from abroad, in my case) difference, let's say between  shoes and some kind of old school skate shoes (the fat ones, like )?  P.S as far as I know most dirt BMXers are quite happy with regular skate shoes. I changed permissions of a file (chmod g+w testfile) and running ls -l testfile gives:  -rwxrwxr-x 1 user1 user1 0 2011-01-24 20:36 testfile   I then added a user to that group ("/etc/group" has user1:x:1000:user2 line), but am failing to edit that file as user2. Why is this so? software updater failed to download repository information When I run software updater it says that it failed to download repository information and to check Internet connectivity.  I have Internet connectivity.  How do I fix this?  editing to add content of sources,list:  sorry I can't figure out how to paste in the content of sources.list without making a huge mess.  Some of it previews as really large bold print.  I tried to follow the block quoting and code rules but it is a huge mess anyway. Mirror changes to side of cube across all six sides Which tools and technologies are used to build the Stack Exchange Network? A horizontal line with the same space-filling behaviour as /hfill? Is there a command or package which would allow me to draw a line in the largest available space in the way that \hfill fills up the remaining space in a line?  i.e., a command to draw a line from the end of the text, wherever it happens to fall, to the right edge of the textwidth (leaving space for any additional text indented to be at the end of the line)? Java Hashmap: How to get key from value? Clipboard for copying and pasting files in command line? In Bash, suppose I visit a directory, and then another directory. I would like to copy a file from the first directory to the second directory, but without specifying the long pathnames of them. Is it possible?   My temporary solution is to use /tmp as a temporary place to store a copy of the file. cp myfile /tmp when I am in the first directory, and then cp /tmp/myfile . when I am in the second directory. But I may check if the file will overwrite anything in /tmp.  Is there something similar to a clipboard for copying and pasting a file?