Prove that $\sin(\pi z)$ can be written as infinite product Prove that  \begin{align} \sin(\pi z) = \pi z \prod_{n=1}^{\infty} \left( 1-\frac{z^2}{n^2}\right) \, \, \, \, \forall \, z \in \mathbb{C} \end{align}  The hint I had it's to use the Fourier series, but I really don't see how.  Any suggestions? Thanks in advance!
Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$ How to prove the following product? $$\frac{\sin(x)}{x}= \left(1+\frac{x}{\pi}\right) \left(1-\frac{x}{\pi}\right) \left(1+\frac{x}{2\pi}\right) \left(1-\frac{x}{2\pi}\right) \left(1+\frac{x}{3\pi}\right) \left(1-\frac{x}{3\pi}\right)\cdots$$
Hogsmeade has a lot of individuals who live there, and it has the distinction of being the only non-muggle community in the UK. Are there any students at Hogwarts who live there during the summer?  I ask because  explicitly says that students must ride the train or not go to the school at all, which seems silly for someone who already lives at the destination.     Many pure-blood families were outraged at the idea of their children using Muggle transport, which they claimed was unsafe, insanitary and demeaning; however, as the Ministry decreed that students either rode the train or did not attend school, the objections were swiftly silenced. I've setup a HTTPS and FTP site in IIS but how can I add users to use the FTP site?  Does they necessarily need an account with password in the computer to be able to use it? Why did Mann's base explode? After the scuffle between Mann (Damon) and Cooper (McConaughey) in Interstellar, our protagonist gets a hold of a radio and alerts his shipmates to Mann's duplicity. Then his ship explodes. What happened there? Are both try to save the file and try saving the file grammatically correct? If so, is there any difference in meaning? Show that the quotient space is homeomorphic to the n-disc SqlServer: can one table's PK be the FK of two table's? I have three tables project, users and proj_user. The PK of proj_user table is the FK of project and users table's pk e.g.(project_id,user_id)  My question is, is it possible if the project_id and user_id's data is different  like project_id = '12345' and user_id = 'abc987'  Please reply  kind regards Vivek I am new to Force.com and trying to learn some concepts.  Currently I am using Single Email Message to send my emails, but I need to change that to mass email, but I don't quite understand how to do that.   I have also created a Visualforce email template, which I would like to use, but it doesn't seem to work.   It would be great if someone could help me out.  //email_disease_map contains pairs of email address and disease ID for (Id id : email_disease_map.keySet())   {     Messaging.SingleEmailMessage mail = new Messaging.SingleEmailMessage();     String[] defaultAddresses = new String[] {'id@test.com'};      List&lt;String&gt; toAddresses = new List &lt;String&gt; (email_disease_map.get(id));      if (!toAddresses.isEmpty())       {         //set addresses for email to be sent         mail.setToAddresses(defaultAddresses);         mail.setCcAddresses(toAddresses);         mail.setTemplateId(Notification);         //set up remaining email         mail.setReplyTo('test@mail.com');          mail.setSenderDisplayName('Ms Test');          Messaging.sendEmail(new Messaging.SingleEmailMessage[] { mail });       }  } Is there a VB.NET equivalent for C#'s '??' operator? Is there a VB.NET equivalent for C#'s ?? operator? Filling triangle doesn't seem to work with PGF/TikZ I was wondering if anyone had any clue on why my triangles are not filled in red within the following piece of code:  \documentclass{article} \usepackage{tikz} \usetikzlibrary{calc,3d}  \usepackage[active, pdftex, tightpage]{preview} \PreviewEnvironment[{[]}]{tikzpicture} \newcounter{number}  \begin{document} \begin{tikzpicture}[x  = {(-0.5cm,-0.5cm)},                     y  = {(0.9659cm,-0.25882cm)},                     z  = {(0cm,1cm)}] \begin{scope}[canvas is yx plane at z=0] \foreach \x / \y / \label in {1 / 0.4 / 1, 0.9 / 0.4 / 0, 1 / 0.5 / 1, 0.7 / 0.3 / 0, 0.8 / 0.4 / 0, 0.9 / 0.5 / 0, 1 / 0.6 / 1, 0.7 / 0.4 / 0, 0.4 / 0.4 / 0, 0.6 / 0.4 / 0, 0.5 / 0.4 / 0, 0.8 / 0.5 / 0, 0.9 / 0.6 / 0, 1 / 0.7 / 1, 0.7 / 0.5 / 0, 0.5 / 0.5 / 0, 0.4 / 0.5 / 0, 0.6 / 0.5 / 0, 0.8 / 0.6 / 0, 0.9 / 0.7 / 0, 1 / 0.8 / 1, 0.7 / 0.6 / 0, 0.6 / 0.6 / 0, 0.5 / 0.6 / 0} { \addtocounter{number}{1} \pgfmathparse{\value{number}}\let\k\pgfmathresult \draw (8 * \x, 8 * \y) node[inner sep=0, outer sep = -0.1pt,color=black,fill,circle] (node-\k) {}; };  \foreach \x / \y / \z / \trash in { 4/5/8/0, 9/11/16/0, 9/16/17/0, 11/10/18/0, 11/18/16/0, 10/8/15/0, 10/15/18/0, 8/5/12/0, 8/12/15/0, 5/2/6/0, 5/6/12/0, 2/1/3/0, 2/3/6/0, 17/16/24/0, 16/18/23/0, 16/23/24/0, 18/15/22/0, 18/22/23/0, 15/12/19/0, 15/19/22/0, 12/6/13/0, 12/13/19/0, 6/3/7/0} { \draw[fill=red]  (node-\x) --  (node-\y) --  (node-\z) -- cycle; } \end{scope} \end{tikzpicture} \end{document}) When handling multiple white-on-transparent png images, I am hindered by the fact that the thumbnails of these images are all-white. Both Adobe Bridge CS6 and Windows Explorer suffer the same problem, I can't discern between two of these pngs.  Is there a way to make these thumbnails useful? Is there some setting I missed that allows changing the thumbnail background colour for a png -- now a deafult white? Can words like "what" be the subject of a sentence? In a question like "Who hears a noise?",  is the subject of the sentence who?  I can think of a few tests for subjects like: "the subject is the phrase that inverts with the auxiliary to form a question". But this is a question and there is no auxiliary. We could say that's because it's a subject, but that would be just to presuppose that it's a subject in the first place.  I can think of tests like: "the form of the verb will agree with the number of the subject", but on the other hand who does not seem to have  fixed grammatical number, and it sometimes seems like who might actually be agreeing with noun phrases later in the sentence:   Who are the most prolific writers of our age? Who is the most prolific writer of our age?   There are also tests like "the subject is usually the first noun phrase in the sentence". But of course if we have a question like:   Who have you bitten?   ... you, not who seems to be the subject. The first noun phrase test is not very good.  I also know that some theories of grammar say that there is a gap in wh- questions that the wh- word is extracted from. So can this gap for instance be the real object of a question? If so, is [See community wiki post in the linked to question]  How can we show whether who is really the subject of my original example question? And how about other wh- words like what? Can they (also) function as subjects? Let $a_1,a_2,a_3,\ldots$ be reals. Prove that if $\sum_{n=1}^{\infty} a_n^{2}$ converges, then so does $\sum_{n=1}^{\infty} \frac {a_n}{n}$.   For this I have shown the case for when $ a_n^{2} \le\frac {|a_n|}{n}$ $\Rightarrow$ $ |a_n|\le\frac {1}{n}$ $\Rightarrow$ $\frac {|a_n|}{n} \le \frac{1}{n^{2}}$ and we know that $\sum_{n=1}^{\infty} \frac {1}{n^{2}}$ converges and hence $\sum_{n=1}^{\infty}\frac {a_n}{n}$ converges by the comparison test. Now considering $ a_n^{2} \ge\frac {|a_n|}{n}$ $\Rightarrow$ $\frac {|a_n|}{n} \le a_n^{2}$ $\rightarrow$ combining the two cases for any n we have: $\frac {|a_n|}{n}\le\frac{1}{n^{2}}+a_n^{2}$ Hence using the comparion test again we know that $\sum_{n=1}^{\infty} a_n^{2}$ converges and $\sum_{n=1}^{\infty} \frac {1}{n^{2}}$ converges hence the sum converges so we can conclude that $\sum_{n=1}^{\infty} \frac {a_n}{n}$ is absoluetly convergent $\Rightarrow$ convergent.  Not to sure if this is correct, any help would be much appreciated, many thanks. Trying to show that $\ln(x) = \lim_{n\to\infty} n(x^{1/n} -1)$ I have a sentence that contains a number of long dataset names. I would like to allow LaTeX to break these names at certain positions that I would like to specify somehow just like I can use \- to allow a word break with hyphenation. But: I would like LaTeX not to insert a dash (which could be mistaken as belonging to the dataset name). How can I do this?   (Something similar is realized in the url package so I guess I could just read that code :) I would not like to use the url package here to be able to make a syntactical (and typographic) difference between urls and datset names.) Since asking for tutorials, videos, resources, links to other content or downloads is considered off topic here, this is a specifically created Community Wiki which gathers resources for Blender and it has been approved by the .  Just write in the appropriate answer/section. If you have concerns, questions, post a  question, so we don't clutter the comments, but you can link your meta question from the comments.  Questions regarding such resources are not allowed anymore, except for very specific and on topic requests (ask on  if you're unsure about your question). Follow the instructions made in the question about how to post, what can be posted, etc.  Learning             Assets    (also see: )   Reference Images/Blueprints (also see: )    Tools    (also see: )      Other   Special Interest ()     Also see the . How to unlock gungeoneers How do I remove 'Places' entries from the Nautilus sidebar? Center-align the verbatim environment Connect to SQL Server with Windows Authentication in a different domain What is a good asymptotic for $f_n = f_{n-1}+\ln(f_{n-1})$?