How do you evaluate a sum of the form $\sum\limits_{n=0}^\infty nk^n$, for $|k|&lt;1$?  If we take $k=\frac{1}{2}$, the sum is $\frac{1}{2} + \frac{2}{4} + \frac{3}{8} + \frac{4}{16} + \dots$, which isn't like any series I know. Wolfram Alpha evaluates this to $\frac{k}{(k-1)^2}$, but I don't know how it finds that.
How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to , but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.  In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$
I'm trying to identify a science fiction series on UK TV during the 70s, but I can only recall the following:   A party of adults and children. There was a woman with a telepathic bond to her cat: she could look into its eyes to see what it had seen. Time travel was involved. I think it may have been inadvertent or uncontrolled time travel. On one occasion the party encountered an Elizabethan seaman who demanded to know if Elizabeth was still queen. They party replied "err... yes" because they didn't want to have to explain time travel. Why is there a clear mirror image off a rough surface at shallow angles? The mirrors we use everyday have a highly reflective smooth surface, so that we can see a clear image.  But I found the same happens on a rough (not so smooth) surface, though only at shallow angles.  For example    this is the logo of physics stackexchange, reflected from the back of an ipad.  Considering that aluminum may contributed to reflection, I tried a different material:    this time a piece of printer paper.  Though not as clear, there is still some visible pattern resembling the logo.  The same effect can be observed on highways, where distant vehicles have a mirror image off the road; I do not have a picture at hand however.  So, why is this happening, and why only on shallow angles? Is there a better way of getting this result? This function fails if num has more digits than digits, and I feel like it should be in the library somewhere (like Integer.toString(x,"%3d") or something)  static String intToString(int num, int digits) {     StringBuffer s = new StringBuffer(digits);     int zeroes = digits - (int) (Math.log(num) / Math.log(10)) - 1;      for (int i = 0; i &lt; zeroes; i++) {         s.append(0);     }     return s.append(num).toString(); } How can I include the cursor in the screenshots created when PrtScr is pressed? Hydraulic Disk brake piston not fully retracting My front disk brake's piston doesn't fully retract.  While I can ride the bike, the brake is slightly rubbing.  (IE, if I just spin the wheel with my hand, it'll go around about one and a half times and then stop.) It's definitely slowing me down.   If this fixable?  The advice I got was that it's not worth the trouble and to buy a replacement set.  The current brakes are Avid Juicy 7s that are about 7 years old Why did I apparently get +97 for joining a site? Don't worry! I got all my 100 reputation :) It just didn't appear that way...    The notification says +100, but why does it say +97 at the top?  Edit  I created an account on server fault, and this time it says +96. I assume it's because of the down votes on this question; however, I thought down votes aren't shown in the rep you gain. I am wondering whether there is any difference between a gerund acting as subject and an infinitive acting as a subject. Why do professional print houses use gamut-limiting CMYK? I recently had some business cards printed, and was warned to use the CMYK colour space if providing TIF files.  When I tried saving from Paint Shop Pro to a CMYK, of course the colours came out very much duller, so I provided a different file type that preserved my colours using RGB. To my horror, the cards came back nevertheless converted to the CMYK dull shades.    A few months previously I had been in a hurry and had printed my own one-sided quasi business cards using glossy photo paper and a cheap home printer.  This reproduced my desired colours perfectly.  Since the whole point of using professional printers is quality (though I admit bulk as well), why do they not use printers that can cope with the RGB colour range?   FYI: the main colour in question, which I chose very carefully, is a slightly yellow-tinged lawn green, which I understand doesn't feature in the CMYK gamut.  So why use that gamut, and is there anything I can do to overcome this issue and get cards printed in the colour I have chosen? Why are all known distributions unimodal? What is Chromatic Aberration? What is Chromatic Aberration? Is it a physical part of the lens, or just an optical illusion? Uncertainty principle and multiple observers I've been modeling a hermit crab and I don't know why but some areas of the mesh are really dark on some spots, like the legs and the claws.   I hope you can help me fix it. Basic buoyancy question: Man in a boat with a stone This comes from a brain teaser but I'm not sure I can solve it:     You are in a rowing boat on a lake. A large heavy rock is also in the boat. You heave the rock overboard. It sinks to the bottom of the lake. What happens to the water level in the lake? Does it rise, fall or stay the same?   I'd say that the water level drops, because when you drop the stone in the lake, the water level rises according to the volume of the stone, BUT the water level decreases by a volume of water weighting the stone's weight, when you take it off the boat.  Is that correct? I can ask any of:     Do you know a breakfast place nearby?   Do you know a nearby breakfast place?   Do you know a good breakfast place?   but I really can't ask:     Do you know a breakfast place good?   Is there a general rule for determining whether an adjective must come before the noun or may come, Spanish-style, after it? Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$ Ubuntu 18.10: visual artifacts (black bar) in dock if extend-height is disabled I am using the default Ubuntu dock with bottom placement and org.gnome.shell.extensions.dash-to-dock extend-height set to false. With Ubuntu 18.10 this produces a noticeable visual glitch, note the black empty spaces at the edges of the dock:    I am getting this both on an upgraded system and on a clean install inside a VM. This was working fine in Ubuntu 18.04. Also happens if the theme is set to something else than Yaru (the default theme in Ubuntu 18.10). Reboot does not help. Dock looks OK if extend-height is enabled.  Any ideas on how to fix this? Change post permalink to external URL from custom field Maximum and Minimum values for ints Say I'm flying from Sydney, to Los Angeles (S2LA), back to Sydney (LA2S).  During S2LA, travelling with the rotation of the earth, would the flight time be longer than LA2S on account of Los Angeles turning/moving away from our position?  Or, in the opposite direction, would the flight to Sydney be faster since the Earth turns underneath us and moves Sydney closer?  ===   Please ignore jet stream effects and all other variables; this is a control case in an ideal environment. By "dramatically" I suppose I mean a delay of 1 hour or more. Remove sudo privileges from a user (without deleting the user)