If we allow some $g(x) = \frac{\sin(x)}{x}$ such that $g(0) = 1$ and also some $f(x)$ where:  $$f(x) = \prod_{k=1}^{\infty} 1 - \frac{x^2}{\pi^2k^2}$$  Then it is said f and g are equal. But my question is, what are the specifics besides the fact that:  Both have the same value at $x=0$. Both have the same roots, and an infinite number of them of course.  I feel like those two criteria definitely don’t justify equality, so I would like to know what legitimately justifies it.
How to prove the following product? $$\frac{\sin(x)}{x}= \left(1+\frac{x}{\pi}\right) \left(1-\frac{x}{\pi}\right) \left(1+\frac{x}{2\pi}\right) \left(1-\frac{x}{2\pi}\right) \left(1+\frac{x}{3\pi}\right) \left(1-\frac{x}{3\pi}\right)\cdots$$
I was peacefully playing GTA Online (yeah, was killing someone coming out of ls customs) then a hacker came and killed me using explosive ammos, I called him a hacker and he teleported to me and gave me 790k. I know that people get banned for receiving this kind of money, what should I do to avoid the ban? Basic Web Development IDE/Editor like Dreamweaver? I've used Windows for many years, using Dreamweaver as my editor as it had a file tree which is based on a project (via Local, FTP or SFTP).  I need a web development IDE with the following functionality:   syntax highlighting for PHP, HTML, CSS, JavaScript  file tree - Local, FTP, SFTP, Project-Based (e.g. via the methods of Local/FTP/SFTP) in other words, like Dreamweaver can open multiple files (tabbed)   Additionally, I wouldn't mind features like debugging, class listings, etc, as long as I can hide them away. So the question is really hard I think. I tried using a simple way by calculating the probability of each combination that makes a sum divisible by six, but it would take forever. Does anyone have any ideas?  Suppose that we roll a six-sided die ten times. What is the probability that the total of all ten rolls is divisible by six? When to use "do" in phrases which are not questions? Example: You do need to drink more water.   or   You need to drink more water.   What is the difference? Since I upgraded to Windows 10, my system has been consuming RAM excessively    I've been reading a bit and determined it's likely a driver leaking memory. So I got myself the Windows Driver Kit and tracked memory usage with poolmon:    However, I don't really know how to proceed from here. Is the item tagged "smNp" the culprit in this issue? How do I go from there to actually identifying the driver?  I tried some stuff like "C:\Windows\System32\drivers>findstr /s smnp ." but it returned no results. I also took a look at the pooltag.txt file and this is the description I found for it:      So yeah, any help would be appreciated. Thanks in advance. Which command / configuration file controls the open file limits on OS X?  Is there a different command for OS X 10.5 / 10.6 / 10.7?  The options I explore below are ulimit, sysctl, and launchctl  "Too many open files" is apparently a common error on Leopard, perhaps other versions of OS X:        There are many (related?) ways to view the open file limits:  $ ulimit -a core file size          (blocks, -c) 0 data seg size           (kbytes, -d) unlimited file size               (blocks, -f) unlimited max locked memory       (kbytes, -l) unlimited max memory size         (kbytes, -m) unlimited open files                      (-n) 2048 pipe size            (512 bytes, -p) 1 stack size              (kbytes, -s) 8192 cpu time               (seconds, -t) unlimited max user processes              (-u) 512 virtual memory          (kbytes, -v) unlimited   $ launchctl limit cpu         unlimited      unlimited       filesize    unlimited      unlimited       data        unlimited      unlimited       stack       8388608        67104768        core        0              unlimited       rss         unlimited      unlimited       memlock     unlimited      unlimited       maxproc     1024           2048            maxfiles    2048           4096         $ sysctl -a | grep files kern.maxfiles = 32768 kern.maxfilesperproc = 16384 kern.maxfiles: 32768 kern.maxfilesperproc: 16384 kern.num_files: 2049   It has been claimed in some of the above posts that these can be modified with the following commands:  sudo launchctl limit maxfiles 16384 32768 sudo ulimit -n 32768 sudo sysctl -w kern.maxfilesperproc=16384 sudo sysctl -w kern.maxfiles=32768   However, out of the above commands, only the sysctl commands have any apparently effect (i.e. ulimit -n and launchctl limit show no change after the above commands have been entered, while sysctl -a does show the requested changes).  The corresponding locations to change these parameters for the OS are:  /etc/sysctl.conf /etc/launchd.conf   I also discovered one  which reads that ulimit only controls the current shell.  How can I adjust upwards the max files / max open files limits on macOS? What is the design pattern for processing command line arguments Let $x_n$ be a sequence of the type described above. It is not monotonic in general, so boundedness won't help. So, it seems as if I should show it's Cauchy. A wrong way to do this would be as follows (I'm on a mobile device, so I can't type absolute values. Bear with me.) $$\left|x_{n+1} - x_n\right| \leq \frac{1}{n^2}.$$ So, we have $$ \left|x_m -x_n\right| \leq \sum_{k=n}^{m} \frac{1}{k^2}$$ which is itself Cauchy, etc., etc. But, of course, I can't just use absolute values like that. One thing I have shown is that $x_n$ is bounded. Inductively, one may show $$ \limsup_{n \to \infty} x_n \leq x_k + \sum_{k=n}^{\infty} \frac{1}{k^2},$$ although I'm not sure this helps or matters at all. Thanks in advance.  Disclaimer I've noticed that asking a large number of questions in quick succession on this site is often frowned upon, especially when little or no effort has been given by the asker. However, I am preparing for a large test in a few days and will be sifting through dozens of problems. Therefore, I may post a couple a day. I will only do so when I have made some initial, meaningful progress. Thanks. Is there a difference between taking a far shot on a 50mm lens and a close shot on a 35mm lens? proof of functional completeness of logical operators If I know that the set of operators {∨, &amp; , ¬} is functionally complete, how do I go about proving/disproving the functional completeness of the following set of operators?  a) $\{\vee,\neg\}$  b) $\{\to,\neg\}$  c) $\{\to\}$  I have looked at the answer here for (b) : Often called "He-Who-Must-Not-Be-Named" and "You know who", was it common knowledge at the Ministry of Magic that Voldemort was the same person as Tom Riddle? As a  developer I'm used to running through constructors:    class Test {     public Test() {         DoSomething();     }      public Test(int count) : this() {         DoSomethingWithCount(count);     }      public Test(int count, string name) : this(count) {         DoSomethingWithName(name);     } }   Is there a way to do this in C++?  I tried calling the Class name and using the 'this' keyword, but both fail. How to test for minimum amount of items in inventory and execute a command if true VirtualBox - how to increase video memory? Correct use of 'with' or 'at' when talking about percentages? I'm writing the following sentence:     Women lead uptake at XX%.   But a friend is saying that it should be written:     Women lead uptake with XX%.   Which one is correct? $M= \{ A \in Mat_{2 \times 2}{\mathbb{R}}| \det(A)=1 \}$ is homeomorphic to $S^{1} \times \mathbb{R}^{2}$ Need to reset the value of sequence in Oracle How does MySQL store data Using Fermats prime numbers to prove that there is infinitely many prime numbers another topology multiple choice Let $S^1 = \{(x, y) \in \Bbb R^2 : x^2 + y^2 = 1 \}.$ Let $D = \{(x, y) \in\Bbb R^2 : x^2 + y^2 \le 1 \}$ and $E = \{(x, y) \in\Bbb R^2 : 2x^2 + 3y^2 \le 1\}$ be also considered as subspaces of $\Bbb R^2.$ Which of the following statements are true?  a. If $f : D \to S^1$ is a continuous mapping, then there exists $x \in S^1$ such that $f(x) = x$.  b. If $f : S^1 \to S^1$ is a continuous mapping, then there exists $x \in S^1$ such that $f(x) = x$.  c. If $f : E \to E$ is a continuous mapping, then there exists $x \in E$ such that $f(x) = x$.