Does the following series converge?  $$\sum \sin(100n) = \sin(100) + \sin(200) + \dots$$
I need to prove that $$\sum_{n=1}^{\infty} {\sin{100n}} \; \text{diverges}$$ I think the best way to do it is to show that $\lim_{n\to \infty}{\sin{100n}}\not=0$. But how do I prove it?
Constants and compiler optimization in C++ I've read all the advice on const-correctness in C++ and that it is important (in part) because it helps the compiler to optimize your code. What I've never seen is a good explanation on how the compiler uses this information to optimize the code, not even the good books go on explaining what happens behind the curtains.    For example, how does the compiler optimize a method that is declared const vs one that isn't but should be. What happens when you introduce mutable variables? Do they affect these optimizations of const methods? I've got a Canon EOS camera that takes a BP-511 battery.  The specs on the battery are 7.4V and 1100mAh.  The battery is dead and I want to buy a replacement, but I'm looking at a third party battery, because it's much cheaper than the Canon branded one.  I've found one with very high reviews online, and it seems identical except that it's rated at 1500mAh instead of 1100mAh.  So the question is: is this difference important to the camera or not? How to back up my entire system? If I want to make a backup of my entire OS (including but not just my home directory), how do I go about this?    Is it as simple as backing up everything in / and then when if I suffer a crash, just copying the files back over? Will this cover grub, and how do I actually do this when the system is inoperable? What's the quickest Schengen consulate to issue a visa in London? Prove that if $k\mid m,$ then $Z_m$ has a subgroup of order $k.$ Prove that if $k\mid m$, then $Z_m$ has a subgroup of order $k.$  Ok, so this doesn't look like too hard of a problem. So do I just show that it is closed under multiplication and inverse? I just don't know how I would do that with a problem like this. Can anyone help me get started please? What equipment and settings do I use for panning shots of moving objects? Does separability imply the Lindelöf property? Does separability imply a sort of Lindelöf property?  Since I can't prove this fact I'm beginning to think that my conjecture is false.  Intuitively, $\mathbb{R}$ has a countable subset $\mathbb{Q}$ which is used to form a countable basis for $\mathbb{R}$ with the usual topology and prove the Lindeöf property. Jointly Complete Sufficient Statistics: Uniform(a, b) Let $\mathbf{X}= (x_1, x_2, \dots x_n)$ be a random sample from the uniform distribution on $(a,b)$, where $a &lt; b$.   Let $Y_1$ and $Y_n$ be the largest and smallest order statistics.  Show that the statistic $(Y_1, Y_n)$ is a jointly complete sufficient statistic for the parameter $\theta  = (a, b)$.     It is no problem for me to show sufficiency using factorization.   Question: How do I show completeness? Preferably I would like a hint.    Attempt: I can show $\mathbb E[g(T(x))] = 0$ implies $g(T(x)) = 0$ for the one parameter uniform distribution, but I am getting stuck on the two parameter uniform distribution.   I tried playing around with $\mathbb E[g(Y_1, Y_n)]$ and using the joint distribution of $Y_1$ and $Y_n$, but I am not sure if I am going in the correct direction, as the calculus is tripping me up. Why is my StackOverflow with an indent and doesn't have reputation Can someone please verify my proof?     Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$?   No. Let $X = \mathbb{R}$.  Clearly, $\{x\} \in \tau_\infty$ for all $x \in X$.  Set $\displaystyle{B = \cup_{x \neq 0} \{x\}}$. Clearly, $X - B = \{0\}$, but $\{0\} \notin \tau_\infty$. Since $\tau_\infty$ is not closed under arbitrary unions, it does not define a topology on $X$. Reshape three column data frame to matrix ("long" to "wide" format) Parent chat user still links to discuss.area51 When viewing a user who has their parent user on chat set to area51, it still links to discuss.area51, instead of the new URL, area51.meta. See:      So can we update these links? Reading and searching long man pages I finally got fed up when wanting to read about bash's read and it's -s option with man bash. I found the right spot eventually (around line 4500), but it was a frustrating as usual, since both /read and even /\s-s\s searches have way too many matches.  So, the question is: How can I read long man pages efficiently, or get same information in other ways, locally? As a specific example, how to reach the relevant documentation after seeing read -s pwd in a shell script? A good answer could be a shell script snippet, or hint about some tool and how it is used, or something else entirely, as long as it helps in finding the right spot to read.  Note: I'm not tagging with  because I want the question to be about man page reading in general, even though that quite possibly is the most commonly encountered humongous man page. TSA broke a lock for which they have a master key. How can I claim compensation? I have bought return tickets from London to Lille Europe on Eurostar. The trains on my outward and inbound journeys both stop at Calais Frethun.  Will Eurostar let me get on in Calais on the inbound? What about getting off at Calais on the outbound leg? I suspect leaving the train early is easier than getting on the train 'late'. how to not commence a command depending on player health? The muffins were rock hard, what did I do wrong? What does // mean in a path? Another layman blackhole question, pulling one end of a string out from behind the event horizon No long explanation is needed,  What would happen if I were to allow one end of a rope to fall past the event horizon of a black hole while I held the other end?  Would I be able to pull it out? Would the rope feel extremely (infinitely?) heavy? Why isn't $\lim \limits_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}$ equal to $1$?