proof of $\sum_{n=1}^\infty n \cdot x^n= \frac{x}{(x-1)^2}$
How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?
Proof or disproof that every integer can be written as the sum of a prime and a square. I'm refreshing some algebra and in the beginning chapters of my book I'm presented with the following simple looking problem that I got stuck on:      Prove or disprove: If $n$ is a positive integer, then $n=p+a^2$, where $a$ is an integer and $p$ is prime or $p=1$.   At first I thought $n=13$ would give a nice counterexample, but then I noticed my book defines primes to include the negative primes. Now $13=-3+4^2$ is a solution.   I am allowed to use the fundamental theorem of arithmetic and the division and euclidean algorithms. I tried to express $n$ and $a$ by their prime factorizations, but I didn't get anywhere from there. I would like a hint which direction to take. Am I just missing something silly? If I have a PATH variable that (when exploded onto multiple lines) contains something like this (with a few notes added by me):  /Users/brandon/.rvm/gems/ruby-2.0.0-p353/bin /Users/brandon/.rvm/gems/ruby-2.0.0-p353@global/bin /Users/brandon/.rvm/rubies/ruby-2.0.0-p353/bin /opt/local/bin /opt/local/sbin /usr/local/sbin /usr/local/bin   # &lt;&lt;--- notice this path appears again later /usr/bin         # &lt;&lt;--- notice this comes AFTER /usr/local/bin /bin /usr/sbin /sbin /usr/local/bin   # &lt;&lt;--- here it is again! /usr/local/git/bin /Users/brandon/bin /Users/brandon/.homebrew/bin /usr/local/mysql/bin /usr/local/share/npm/bin /Applications/Postgres93.app/Contents/MacOS/bin /Users/brandon/.rvm/bin   is it possible for the presence of the second /usr/local/bin/ to cause the /usr/bin to take precedence over /usr/local/bin?  This seems crazy (to me: I've never heard of such a thing) but that's exactly the behavior I'm seeing.  I've just installed PHP with Homebrew, and yet the system PHP (from /usr/bin) is taking precedence:  $ which php /usr/bin/php   BUT, after manually altering the PATH (rather than digging through all the files sourced by my .zshrc to try to figure out why &amp; where the duplication is coming from) to remove the second instance of /usr/local/bin, I find that /usr/local/bin is taking precedence over /usr/bin as I would have expected it to in the first place:  $ which php /usr/local/bin/php   What would cause this?  Is this normal? How to create a GUID/UUID in Python How do I create a GUID in Python that is platform independent? I hear there is a method using ActivePython on Windows but it's Windows only because it uses COM. Is there a method using plain Python? When starring a message from the star board, it's not reflected in the main chat window I entered the USA on a three month visa waiver program from Australia. I am about to return to  Australia and would like to come back again to the USA. How long do I have to wait until I can re-enter the USA for another 3 months? "did shoot" vs "shot" This morning I read this sentence (see ):     On July 24th and again on July 29th,   Egyptian police did shoot dead unarmed   African migrants attempting to cross   that border.   Why "did shoot" and not "shot"? I'm working on a new edition of a book which uses a very unusual style of bibliography: Entries are sorted in author-year fashion, but then numbered per author or group of authors, e.g.:  Milnor, J.:    1. Construction of universal bundles: I, Ann. Math., (2)63:272-284 (1956).    2. Construction of universal bundles: II, Ann. Math., (2)63: 430-436 (1956).    3. On manifolds homeomorphic to the 7-sphere, Ann. Math., (2)64: 399-405       (1956).    4. The theory of characteristic classes, mimeographed, Princeton University,       Princeton, N.J., 1957. Milnor, J., and M. A. Kervaire:    1. Bernoulli numbers, homotopy groups, and a theorem of Rohlin, Proc. Intern.       Congr. Math., 1958. Milnor, J., and E. Spanier:    1. Two remarks on fibre homotopy type, Pacific J. Math., 10: 585-590 (1960).   Citations are formatted as Milnor [3], and so on.  Is there any bibtex style for something strange as this? Or can this type of bibliography be created with biblatex without too much coding? Why do adult insects have 6 legs? Apparently, there is an advantage to having 6 legs in the insect world. What is that advantage, if anything? Why would such an advantage exist for insects, but not for other, larger land animals?  What do those middle legs do that the front and hind legs cannot, if anything? Are they really essential, or do they just generate more muscle power? How to pass parameters to an alias? For bash script, I can use "$@" to access arguments. What's the equivalent when I use an alias? how does the circuit of a basic Variable speed electric drill work? i am trying to 'automate' the control of an electric drill motor that im taking from a garden variety Variable Speed drill (from Harbor Freight)  it has a universal AC motor which I basically undertand and a DPDT switch to reverse direction and a triac citcuit to vary speed    Question 1 . i see 4 wires going to the 'field/Stator coil(s)'. Are there 2 seperate field/stator coils here? is this to provide low/high power.  I think in theory , you only need 1 coil with 2 wires and that coil is wired in series with the brushes... is that right?   how does this circuit work with 2 coils.  Question 2 . how does this variable speed circuit work.  is the wiring diagram correct (it doesnt show a variable resistor and it doesnt show what drives the control input of the traic? why are there two switches in the thing? does switch 2 close at high speed and thus shunt around the triac?  please see the attached 4 diagrams Each Stack Exchange site should provide users the option of using a dark mode design/theme.  Why is this feature missing?  The major IDEs are already a letdown regarding dark color schemes. Which browsers are officially supported, and what else do I need? Why could the pictures and paintings in Hogwarts talk and the photos of Harry's parents couldn't? Field that is a subfield of own of its subfields Let $K$ and $L$ be fields. We have homomorphisms $f: K \to L$ and $g: L \to K$. Are $K$ and $L$ necessarily isomorphic? What should I do if I have two libraries that provide functions with equivalent names? How do I install Wi-Fi drivers for Realtek RTL8723DE device in Ubuntu 16.04?  lspci -v | grep -i network 02:00.0 Network controller: Realtek Semiconductor Co., Ltd. Device d723 Does the everywhere differentiability of $f$ imply it is absolutely continuous on a compact interval? Suppose $f$ is differentiable everywhere on $[0,1]$. Must $f$ be absolutely continuous on $[0,1]$? I know this is true if $f'$ is integrable but I'm not sure in this more general case. Any way to tell apart a CPU defect from a mainboard defect? I have a fairly modern desktop computer, an AMD Athlon II X2 based silent PC, that does not work.  When turned on, it will start physically (all fans are rotating, disks start up...) but not give a signal on any of the graphics ports (DVI, VGA, and HDMI, I tried all three). Also, the reset button does not seem to have any effect.  I have stripped the mainboard bare of all SATA connections, extension cards, and the one 2GB RAM chip to eliminate them as the problem source, but to no avail, so I think it's fair to assume it's either the mainboard or the processor that are at fault.  However, I have neither a replacement mainboard, nor a replacement processor handy to identify which one is broken. The cause of the defect is unknown, so for all I know, it could be both. Therefore, I'm reluctant to buy replacement hardware blindly before knowing more.   Is there any way to further diagnose (or at least get some indication) which component is broken without buying replacement hardware? Shouldn't the Terms of Service use Stack Overflow, Inc.? The  still uses the term Stack Exchange, Inc. However, I believe that the name is now Stack Overflow, Inc. , so I believe that should be changed to that. It is weird to have the wrong name in a legal document.   Yes, I actually read the Terms of Service. You should also do that. How does the EMH hold onto a phaser while having bullets go through him? In the Voyager episode: "Future's End, Part II" the doctor rescues B'elanna and Chakotay.     PORTER: They've got lasers! A black man and some bald guy.  (The EMH   enters. Porter and Butch's bullets go straight through him.)       PORTER: God in heaven help us.       EMH: Divine intervention is unlikely.  (The EMH phaser stuns them,   then unties Torres and Chakotay.)   In an earlier episode of Voyager, "Phage":     PARIS: But a hologram is just a projection of light held in a magnetic   containment field. There's no real matter involved. (The EMH slaps   Paris.)       EMH: Now, you hit me.  (Paris's hand passes through the EMH's head.)       EMH: The magnetic containment field that creates the illusion of my   body can be modulated to allow matter to pass through it or be   stopped. I might be able to modulate the holographic lungs in the same   way, allowing oxygen and carbon dioxide to pass from the lungs to the   blood stream.   In order to make the above adjustments the doctor had to touch several buttons on a control panel. I realize the time I refer to above he was using the holo-emitter instead.    How does it work exactly that the doctor can have bullets pass straight through him but he's able to hold on to a phaser without it falling out of his hand? Can he simultaneously have part of his body be transparent and the rest be more like matter at the same time? What if the bullets hit the hand holding the phaser? I think one had a shotgun as I recall so there probably would have been pellets going all over the place.