The  includes this summation:  $$\sum_{n=0}^{\infty}{\frac{1}{n \cdot 2^n}} = \ln 2$$  I was very surprised when I saw this, and I don't have a clear sense of where it comes from. This doesn't seem related to the Taylor series for $\ln x$. It somewhat reminds me of the sum  $$\sum_{n=0}^{\infty}{\frac{n}{2^n}}$$  which converges to $2$, but I'm not sure whether there's some way to manipulate the first sum in a way that relates it to the second. It also vaguely reminded me of the sum from the harmonic series, except that the $2^n$ term in the denominator breaks that similarity as well.  Is there a simple explanation as to why this sum gives $\ln 2$? Does this generalize to any other natural logs (say, could a modified sum give back $\ln 3$), or is this oddly specific to $\ln 2$?
What would be the sum of the following series?  $$\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$$  Thanks
Changing an element's ID with jQuery I need to change an element's ID using jQuery.    Apparently these don't work:  jQuery(this).prev("li").attr("id")="newid" jQuery(this).prev("li")="newid"   I found out that I can make it happen with the following code:  jQuery(this).prev("li")show(function() {     this.id="newid"; });   But that doesn't seem right to me. There must be a better way, no? Also, in case there isn't, what other method can I use instead of show/hide or other effects? Obviously I don't want to show/hide or affect the element every time, just to change its ID.  (Yep, I'm a jQuery newbie.)  Edit I can't use classes in this case, I must use IDs. Are Harry Potter and Ginny Weasley Distant Relatives? Is there a way to install a newer version than in the repository?  Ubuntu installs vagrant 1.4.3 at the moment, but I need at least 1.5 How to graphically represent a language Expressing $1.24\overline{123}=1.24\;123\;123\;123\;\ldots$ as the ratio of two integers Huge confusion with Fermions and Bosons and how they relate to total spin of atom I am supremely confused when something has spin or when it does not. For example, atomic Hydrogen has 4 fermions, three quarks to make a proton, and 1 electron. There is an even number of fermions, and each fermion has a 1/2 spin. Since there are an even number of fermions, the total spin value is an integer. This spin number is the "intrinsic" spin number that cannot be changed, but its orientation "up" or "down" can be changed.  For atomic Hydrogen, it is a Boson because it has integer spin, however it also has a single electron. I read on physics forums, , that the spin of atom comes from the electrons and not its nucleus. I also read on here, , that the spin of atomic Hydrogen is 1/2! The answer says atomic Hydrogen has spin 1/2 because it ignores the nuclear spin.  This is one thing that is confusing me. Shouldn't atomic Hydrogen have an integer spin because of the nuclear component? So does atomic Hydrogen have spin and is affected by a magnetic field? Nuclear spins are affected by magnetic fields, but they aren't as affected as electrons according to the discussion on physics forums.  Why do we ignore nuclear spin sometimes? Also, can someone help me out here with all the possibilities?  Is there a Boson with an half integer spin value? (Surely, there must not be) However atomic Hydrogen is one of those cases! (It seems...) (Why don't we cancel out the nuclear spin with the electron spin?)  Say we have another atom that is a Boson, It has unpaired electrons in different orbitals, so what determines whether or not electrons fill in orbitals as spin up or down? Does spin down nuclear spin cancel out a electron up spin? How to set security options for a PDF using pdflatex My questions is very simple. How can I set security setting for a PDF file created with PDFLatex?  I should set some password or print permissions, etc. We have the matrix: $$\left[\begin{array}{rrr} 0.8&amp;0&amp;0\\ 0&amp;1.2&amp;0\\ 0&amp;0&amp;1\\\end{array}\right]$$ Finding the eigenvalues and eigenvectors are easy. But finding ${\bf M}^n$ when $n \rightarrow \infty$ is the hard part. Any suggestions? The symmetric cryptosystem one-time pad (OTP) seems to be very beautiful since it is perfectly secret according to Shannon. Many books, however, point out the main drawback: one must create a secret key the same length as the plaintext. For this reason, the concept of perfect secrecy has these days been abandoned, and we prefer other cryptosystems.   Now, for example, suppose I want to encode my text message with the ASCII scheme, and moreover suppose that this plaintext is 10 gigabytes long, therefore (using an OTP) the key must be the same length, 10 gigabytes. I think, however, this is a ''sustainable price'' because I'm sure that the ciphertext can't be attacked.  Practically, I don't understand why having such long keys with an OTP is a big disadvantage if we reach "the dream"-- namely, perfect secrecy. Nowadays the storage of information is very easy, so what is the real problem? I can, for example, share the "long key" in person. When do I use active vs passive perception? So, I'm trying to figure out how to run the scene from the beginning of Lost Mine of Phandelver (5e starter set) where the party is ambushed by goblins and I'm really confused as to how Passive Wisdom (Perception) works.  The included rulebook mentions on page 6 :     When you hide, there’s a chance someone will notice you even if they aren’t searching. To determine whether such a creature notices you, the DM compares your Dexterity (Stealth) check with that creature’s passive Wisdom (Perception) score, which equals 10 + the creature’s Wisdom modifier, as well as any other bonuses or penalties. If the creature has advantage, add 5. For disadvantage, subtract 5. For example, if a 1st-level character (with a proficiency bonus of +2) has a Wisdom of 15 (a +2 modifier) and proficiency in Perception, he or she has a passive Wisdom (Perception) of 14.   So the party is being ambushed by goblins and at that point the DM guide mentions on page 7 :     Check to see who, if anyone, is surprised. [...] Make a Dexterity (Stealth) check for the goblins, rolling once for all of them. Roll a d20, add the goblins' Stealth skill modifier (+6) to the roll and compare the result to the characters' Passive Wisdom (Perception) scores.   So to me this reads like I should roll 1d20+6 for the goblins and compare that to the static Passive Wisdom (Perception) score on the character sheet of a PC. The PCs don't get to roll anything as they're not actively looking for enemies, even though they are investigating the dead horses.  I would also think that if the characters were supposed to roll, the DM guide would clearly instruct so, given that this is the starter set.  However, watching  at around 8:01 I notice that the DM is explicitly asking his party for a perception roll. And the same thing happens on a couple of other videos where people are running the Lost Mine adventure. The DM always requests a roll.  So what gives? Shall I take it that the DMs ask for the roll because they consider the party to be actively seeking for hidden foes? Or am I just misreading the rulebook and the DM guide? If I have a sentence that starts with additionally, finally, consequently, etc. do I always have to put a comma after it? Or is there a different rule? File1:  .tid.setnr := 1123  .tid.setnr := 3345  .tid.setnr := 5431 .tid.setnr := 89323   File2:  .tid.info := 12 .tid.info := 3 .tid.info := 44 .tid.info := 60   Output file:  .tid.info := 12 .tid.setnr := 1123 .tid.info := 3 .tid.setnr := 3345 .tid.info := 44 .tid.setnr := 5431 .tid.info := 60 .tid.setnr := 89323 The following quote from  assumes "the URL" comes from somewhere and uses it to restrict the Host header. Where does "the URL" come from? Is it from the first line of a HTTP request (e.g. after GET), or from the input to a web client (e.g. an argument to curl, or the input in the address bar of a web browser)? Thanks.     The Host header specifies the Internet host and port number for the   resource being requested, as obtained from the original URL:  Host = "Host" ":" host [ ":" port ]       In particular:      • If the Host header does not contain a port, the default port for the   scheme is assumed.      • If the URL contains an IP address, the Host header should contain   the same address.      • If the URL contains a hostname, the Host header must contain the   same name.      • If the URL contains a hostname, the Host header should not contain   the IP address equivalent to the URL’s hostname, because this will   break virtually hosted servers, which layer multiple virtual sites   over a single IP address.      • If the URL contains a hostname, the Host header should not contain   another alias for this hostname, because this also will break   virtually hosted servers. Why does the zoom sometimes stop at a point? I often find that I can zoom in to a point, but then cannot zoom any more, even when there is a lot of room to zoom to the meshes on the scene.   This is particularly annoying in the following scenario:  I often land up left clicking on the scene and the 3D cursor shifts there. I use Shift+C to shift the 3D cursor back to the origin. However, this also centers the view on the cursor and shifts the view to show all objects, and sometimes this zooms things way out &ndash; at this point, if I try to zoom in, I can zoom in to a point (using the pinch/zoom on trackpad), but then it stops zooming in.   How do I zoom in after that? Is there anything else I am doing wrong in this case? Gmail, how to forward email to group using a filter How to hook into the CPT's title placeholder? Do the laws of physics evolve? Extrude and bevel an imported svg curve Female main character in an ice world earning gems/runes Here's what I remember:   The main character's female (99% sure) There is one part where she has to cross a big tundra and she has to kill a big ice creature The kids in the story go to a school-like place where they train for something. I'm pretty sure they earn gems or runes or something like that There's one part where the main character has to play a piece of music perfectly to earn her gem/rune thing It's definitely a series; young-adult (I read them in middle school)   Thanks! Test whether a glob has any matches in bash