In the following series:  $ \displaystyle\sum\limits_{n=1}^{\infty}n*\frac{1}{2^n}$  I've found that the series converges to 2 by looking it up but how would one calculate the summation? You can't use the formula for a geometric sum because the series' terms don't differ by a common ratio. I'm sure I'm just missing something but it's been a while since I've done series. Thanks!
I have put his on wolfram and obtained answer as follows:  $\sum_{n=1}^{\infty}\frac{n}{2^n} = 2$  And the series is convergent too because $\lim_{n\to\infty} \frac {n}{2^n} = 0$  However I am wondering if there is a convenient way to solve this; I don't think you can represent it by a geometric progression either. So how do we have to do it on paper?
Prove that if $a+b+c$ divides $abc$, then $a+b+c$ must be composite. I have seen another post for this same question but it never got an answer. Since I am fairly new, I can't comment on other's posts, therefore I am creating a new one. The question states:  If a, b, and c are positive integers, prove that if $(a + b + c) | (a*b*c)$, then $a+b+c$ is composite.  I do not really know how to approach this.  The link to the post with the same question is: What role does "spontaneously symmetry breaking" played in the "Higgs Mechanism"? In talking about Higgs mechanism, the first part is always some introduction to the concept of spontaneously  symmetry breaking (SSB), some people saying that Higgs mechanism is the results of SSB of local gauge symmetry, some people says that we can formulate Higgs mechanism in a gauge invariant way, some people also says that we need only a non-zero vaccum expectation value... I am confused about this different or maybe same point of views.  In this post:  , the most highly voted answer, I still can't feel how SSB worked in Higgs mechanism. It seem that the validity of last part, the appearance of a mass term for $A$, is guaranteed if we have a non-zero equilibrium value $\phi_0$ to expand around. I do not see that the requirement that the phase of the field $\phi$ need to be fixed at some particular value to generate mass term. Thus it seems to me it is not true that SSB is really indispensable for Higgs mechanism.  To put it simply:  The spontanously breaking of what is attributed to Higgs mechanism?   local gauge symmetry global symmetry, since breaking of a "gauge symmetry" should not have any effect on physics. In higgs mechanism, the really broken symmetry is a global one. Mathematically, it is similar in looking as fixing a gauge, but one should not think it as a spontanously breakdown of local gauge symmetry. other.   Is SSB really indispensable for Higgs mechanism?    yes, Higgs mechanism is relied on the SSB of some symmetry (above question), the other approches of description eventually has spontanously broke some symmetry. No, the SSB is just one way to describe Higgs mechanism (or even not a complete way), what is really need is the non-zero vaccum expectation value, for example in the linked post the requirement for the mass term to occur is to have some non-zero expectation value of $\phi$ to expand around, we do not need the phase of the field to be fixed, thus the symmetry is not broken. Other.   some reference materials:    States that SSB of local gauge symmetry is impossible.  in the abstract states that:       gauge symmetries merely reflect a redundancy in the state description   and therefore the spontaneous breaking can not be an essential   ingredient. Indeed, as already shown by Higgs and Kibble, the   mechanism can be explained in terms of gauge invariant variables,   without invoking spontaneous symmetry breaking     In the introduction it says:      In particular, we emphasize that global U(1) phase rotation symmetry,   and not gauge symmetry, is spontaneously violated, and show that the   BCS wave function is, contrary to claims in the literature, fully   gauge invariant How to find all the available tools in mac terminal? I am trying to setup a bunch of development projects in my Mac. I have to run various tools like python, ruby, scala, groovy etc.  To see if a tool is available I use options like   ruby --version.    Instead I would like to see all the list of tools available in Mac terminal via a single command. What is the command/options for it? Conjectures that have been disproved with extremely large counterexamples? I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.  I'm sure that everyone here is familiar with it; it describes an operation on a natural number – $n/2$ if it is even, $3n+1$ if it is odd.  The conjecture states that if this operation is repeated, all numbers will eventually wind up at $1$ (or rather, in an infinite loop of $1-4-2-1-4-2-1$).  I fired up Python and ran a quick test on this for all numbers up to $5.76 \times 10^{18}$ (using the powers of cloud computing and dynamic programming magic).  Which is millions of millions of millions. And all of them eventually ended up at $1$.  Surely I am close to testing every natural number? How many natural numbers could there be?  Surely not much more than millions of millions of millions. (I kid.)  I explained this to my friend, who told me, "Why would numbers suddenly get different at a certain point?  Wouldn't they all be expected to behave the same?"  To which I said, "No, you are wrong!  In fact, I am sure there are many conjectures which have been disproved by counterexamples that are extremely large!"  And he said, "It is my conjecture that there are none! (and if any, they are rare)".  Please help me, smart math people.  Can you provide a counterexample to his conjecture?  Perhaps, more convincingly, several?  I've only managed to find one! (Polya's conjecture). One, out of the many thousands (I presume) of conjectures. It's also one that is hard to explain the finer points to the layman. Are there any more famous or accessible examples? Mismatch Reputation in the thumbnail and the actual Reputation Black screen after installation of Ubuntu 18.04 A proof of the fact that the Fourier transform is not surjective from $\mathcal{L}^1(\mathbb{R})$ to $C_0( \mathbb{R})$ Let $f_n = \mathbb 1_{[-n,n]}$ for all $n \in \mathbb{N}$   Compute explicitly $f_n \star f_1$ for all $n \in \mathbb{N}$.  Show that $f_n \star f_1$ is the Fourier transform of $g_n = \frac{  \sin{(2\pi x)} \sin{(2 \pi nx)}}{\pi ^2 x^2}$  Show that $\| g_n \|_1 \to \infty$  Deduce that the Fourier transform is not surjective from $\mathcal{L}^1(\mathbb{R})$ to $C_0( \mathbb{R})$    Here, $C_0(\mathbb{R})$ is the space of continuous functions $f : \mathbb{R} \to \mathbb{R}$ that tends to $0$ at $-\infty$ and $+\infty$. I managed to compute 1) and proved 2). But I don't know how to prove 3). Moreover, I don't see how can 1), 2) and 3) prove 4). Can anyone help? How to hide "internal" polygon boundaries? Add high-resolution-graphics and / or .svg-icons Fourier series identity Is there a way to combine custom settings with custom labels ? So that the output from a setting value would be a custom label ?   To explain:  In example, we have an international organisation, and say countries are stored as an ISO code. As this is a limited and stable set of data, using custom settings would be the appropriate way to deal with say the full country name.   The issue arises when I do not just want to store the English country name, but want to have it translatable, presenting the right translation depending on the user's locale. Now, this is the point where I'd like to integrate custom settings with custom labels.  So that say the custom setting with name 'some iso code' doesn't return the full country name, but Label.Country'some iso code', which is than automatically translated in the right version.   Sounds good right. But as custom settings are fixed to certain types, as far as I've gotten is put the label name in the settings as a string value. And then .. i'm stuck :).  No way to cast a string to a Custom label, or have it be interpreted (and translated) as such, or is there ?   I'm aware this can be done in a custom object &amp; apex, I was hoping for a more automated, "cleaner", less resource consuming way, better integrated, configurable way.  Additional clarrification: *The goal is to extend the use of custom labels (translation workbench), by adding a mapping from any data-based value to a custom label.*   While we could do a long matching process in apex, it would be faster &amp; neater if the custom settings mechanism could be used. There are disadvantaged to custom objects, they require soql &amp; apex, while custom settings &amp; custom labels are cached. They too can be used in many other places. Countries were an example, I'd use this mechanism for every variable value that needs to be translatable. I assume the difference in the number of Stack Exchange sites/communities is a mistake... correct? Peppers evolved chemicals like capsaicin to discourage animals from eating them, and that chemical itself can cause internal damage if consumed in large quantities, though this is usually only a concern for the hottest of peppers (which people do eat).   Still, why is it that humans seem to be the only animal that often "like" spicy food? Thousands of .log files in Exchange 2007 Server folder I'm a bit of an Exchange admin noob so I don't know if this is normal, but on our Exchange 2007 server under C:\Program Files\Microsoft\Exchange Server\Mailbox\First Storage Group\ there are 30,000 1MB .log files.  I don't know if those should be there or not, but they keep multiplying like little jackrabbits.  I've so far resisted deleting them as I'm sure something would break, but it's come to the point now where our 80GB disk is completely full up and Exchange has stopped receiving messages.  My best guess is that these are transaction logs and would be useful in rebuilding the database should something go awry, but 30GB of transaction logging seems a bit excessive.  Is there a way to limit it or is that even a good idea?  And of course did I completely miss the point? I want to advance in the ranks of Senator Lavigny-Duval. How can I get merit points? I know I can participate in expansion goals, but is there a better way than that? I want to change the default location of "Save a screenshot to Pictures" to save it to Desktop.   If that is not possible, is there any other way to do it? I am using Ubuntu GNOME 17.04.  NOTE: I don't want to change the location with dconf-editor as it changes the location of screenshot applet and not of the shortcut I require. What are the reputation requirements for privileges on sites, and how do they differ per site? Formerly Reputation requirements compared. What are the reputation requirements for privileges on the various sites in the network? Also, how do reputation requirements for various privileges compare on different Stack Exchange sites? Specifically, the following types of sites:  Designed sites e.g. Stack Overflow, Ask Ubuntu, Gaming, etc. Public beta and non-designed sites Private beta sites  ...as well as any site-specific differences. Custom thumbnailers don’t work on Ubuntu 18.10 and 18.04 Recently I had a bunch of my own scripts for the thumbnails in Nautilus: , , office documents and so on.  Seems like either I’m doing something wrong (thumbnail generation behavior changed), or custom thumbnailes don’t work in latest GNOME at all, even my previous scripts, which worked smoothly on previous versions of Ubuntu.  I’ve done some research, tried to run a simple script (see below) and it didn’t work out.  My test thumbnailer in /usr/share/thumbnailers/z.thumbnailer:  [Thumbnailer Entry] Exec=/home/maximal/thumb.sh %s %i %u %o MimeType=image/webp;image/x-webp   Where /home/maximal/thumb.sh is:  #!/bin/bash echo $0 &gt;&gt; /home/maximal/thumb.log echo $1 &gt;&gt; /home/maximal/thumb.log echo $2 &gt;&gt; /home/maximal/thumb.log echo $3 &gt;&gt; /home/maximal/thumb.log echo $4 &gt;&gt; /home/maximal/thumb.log   Then, when I remove thumbnail cache rm -rf ~/.cache/thumbnails and restart Nautilus nautilus -q, my log file doesn’t contain anything and ~/.cache/thumbnails/fail/gnome-thumbnail-factory contains all the failed thumbs (looks like corrupted/empty PNGs):    So, apparently, Nautilus even didn’t try to execute my thumbnailer script. When I try to run thumb command itself, it works perfectly. For example convert through cwebp|dwebp package:  convert -thumbnail 256x256 file.webp png:file.webp.png   When I wrap this command to a thumbnailer, it looks like it wasn’t even requested to run by Nautilus.  [Thumbnailer Entry] Exec=convert -thumbnail %sx%s %i png:%o MimeType=image/webp;image/x-webp;   Exactly the same situation occurs with other my thumbnailers: audio waveforms, office documents, photoshop files and so on.  Could anyone help? Select top 10 records for each category I want to return top 10 records from each section in one query. Can anyone help with how to do it?  Section is one of the columns in the table.  Database is SQL Server 2005. I want to return the top 10 by date entered.  Sections are business, local, and feature.  For one particular date I want only the top (10) business rows (most recent entry), the top (10) local rows, and the top (10) features. I am new to blender (yes, laugh at the obligatory doughnut) and I when I set up the camera (I use NUM 0 to access the camera and move it into position with &quot;Lock camera to view&quot;) in the viewport it shows as the first attached image, but when I render it the image is further back as shown in the second image. Is this normal behavior or am I missing something?