Evaluating $\sum\limits_{k=0}^n\cos(kx)$ and $\sum\limits_{k=0}^n\sin(kx)$ without Complex Numbers Alright, so the standard way to evaluate $\sum\limits_{k=0}^n\cos(kx)$ and $\sum\limits_{k=0}^n\sin(kx)$, is to respectively take the real and imaginary part of  $$\sum_{k=0}^n{\rm e}^{ikx}={\frac {{{\rm e}^{i \left( n+1 \right) x}}-1}{{{\rm e}^{ix}}-1}}$$ $$=\frac12\left({\frac {\cos \left(   \left( n+1 \right) x \right) -\cos \left( nx  \right) }{\cos \left( x \right) -1}}+1 \right) +\frac12i\left( {\frac {\sin \left(  \left( n+1 \right) x \right) -\sin \left( nx  \right) -\sin \left( x \right) }{\cos \left( x \right) -1}} \right)$$  While this is relatively simple concept and easy to remember how to derive, is there a way to come up with these results that do not rely on complex numbers? Also, do not give an inductive proof, as that is not particularly constructive, and the form of the sum values is not easy to determine by inspecting.
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series:  $$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl( \frac{ 2 a + (n-1)\cdot d}{2}\biggr)$$  There is a slight difference in case of $\sin$, which is: $$\sum_{k=0}^{n-1}\sin (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \sin\biggl( \frac{2  a + (n-1)\cdot d}{2}\biggr)$$   How do we prove the above two identities?
Free, open-source substitutes for Mendeley? is (mainly) a proprietary social network to share basic citation data and research papers.   Are there any free and open source substitutes for this service?   A technical possibility: users could collect BibTeX data and a hash and share this information.  It would be very useful to have such a free service because every journal provides the citation data in a different format. That includes false field entries, broken files and hidden download buttons on the website. Is the following statement provable? How do I determine if a number is prime?  I'm writing a program where a user inputs any integer and from that the program determines whether the number is prime, but how do I go about that? Pokemon Go login issues after update I downloaded the July 30th update for Pokemon Go and when I open the app I'm prompted to login, But once I login, it redirects me to this page on Cookies::     If I click "Done" in the top left, it prompts me to login again! I've tried reinstalling the game, restarting my phone, etc. I can't much enable Cookies on the inline Safari browser. It doesn't make sense, and cookies are on in Safari (because I don't use Safari and they're enabled by default).  How can I fix this? To be clear on this, I know what is the definition of an inner product space and some properties and theorems about them. What I am asking for is an intuition for this definition in the complex case. In the real case, the intuition (or at least one of them) is  geometric: The inner product of two vectors is the length of the projection of the first to the second scaled by the norms of both vectors so that it is symmetric (modulo some details). In particular I (and everybody else) think of "inner product zero" as geometric orthogonality and of orthonormal bases as, well, orthonormal bases and so on. The question is, what should I think about when working with complex (or should I say hermitian?) inner product spaces? what is the "meaning" of the complex number associated to two vectors called their inner product?  I will be happy to hear all kinds of answers. For example, what physical phenomena does it model or in what mathematical situations does in "naturally" appear. Answers that stress the "nice structure" resulting are also welcome, yet I feel that by itself  it is a bit unsatisfying. Almost sure convergence of the Poisson process Let $N = \{N(t) \}_{t\geq 0 }$ be a Poisson process. I already know that $N(t)- \lambda t$ is a martingale where $\mathbb{E} [ N(t) ] = \lambda t$.   I want to prove that $$ \frac{N(t)}{t} \rightarrow \lambda, \quad t \rightarrow \infty \quad \text{ a.s.}. $$  To prove this I received a hint that I should use the martingale law of large numbers which states that:  Theorem: (Martingale LLN)  Let $S= (S_n = \sum_{k=1}^n X_k)_{n=1,2,\ldots}$ be a martingale with respect to the filtration $(\mathcal{F}_n)_{n=1,2,\ldots}$. Then $\sum_{k=1}^n X_k/k$ converges almost surely on the set $\left\{ \sum_{k=1}^\infty k^{-2} \mathbb{E}\left[ X_k^2 \mid \mathcal{F}_{k-1} \right] &lt; \infty \right\}$. Hence $S_n/n \rightarrow 0$ a.s. on the same set.  Now to use this theorem I think I need to use Doob-Meyer to decompose $N(t)$ as the sum of a martingale and an increasing process. Which would be the martingale $M(t) = N(t) - \lambda t$ and increasing process $A(t) = \lambda t$. But how is $M(t)$ the sum of a martingale? And I do not know how this exactly relates to the thing I have to prove. Thanks for any help. What is scientific mind? How do I iterate over a range of numbers defined by variables in Bash? How do I iterate over a range of numbers in Bash when the range is given by a variable?  I know I can do this (called "sequence expression" in the Bash ):   for i in {1..5}; do echo $i; done   Which gives:     1    2    3    4    5   Yet, how can I replace either of the range endpoints with a variable? This doesn't work:  END=5 for i in {1..$END}; do echo $i; done   Which prints:     {1..5} The Help Vampire problem pm-hibernate freezes and/or fails to resume on Dell XPS 13, 14.04, 3.19 kernel Facebook message notification showing blank message Since I updated the system to 10.11.5, the Facebook message notification is not showing message text.   Is this a bug in this software upgrade of OSX? I am using Ubuntu 11.10 and my screensaver keeps coming on while I'm watching a movie.  I don't remember this happening in 11.04 and can't remember how to stop it? E2099 Overflow in conversion or arithmetic operation Calculate $ \int_{-2}^{0} \frac{x}{\sqrt{e^x+(x+2)^2}} dx $ $ \int_{-2}^{0} \frac{x}{\sqrt{e^x+(x+2)^2}} dx $ = -$  2.887270... $  The function has no antiderivate and there's no symmetry here to help, if you solve this I would be thankful if you would for a highschooler ^^ Extra slash added to the chat room URLs in the side bar There is a extra slash added to the chat room URLs in the side bar. This is not causing any redirection issue or 404 not found error. Just observed it and want to notify to the development team about this. Seems it is not a bug, but that can . Currently the following URLs are binded against the chat rooms in the side bar.  Tavern on the Meta https://chat.meta.stackexchange.com//rooms/89  The Fire Department https://chat.meta.stackexchange.com//rooms/1181   GIF for the reference: 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? Will moving blog posts from under an existing category to a new SUBcategory have a negative impact from an SEO POV? Example where Hamiltonian $H \neq T+V=E$, but $E=T+V$ is conserved I'm looking for an example of a Hamiltonian $H$, where $H\neq T+V$, but the total energy in the system, $E=T+V$, is still conserved.   While I'm at it, I might as well add that I'd be most interested in an example from a classical field theory. Furthermore, I am looking for a nontrivial example, e.g. where $H$ doesn't just differ from $E$ by some gradient function that integrates to 0 in the action.   In particular, I am trying to get a better understanding of the physical implications of $H$ being conserved when nontrivial variables are used, i.e. physically, what does $H$ this correspond to. Conversely, can one have a system where $E = H$ in one frame, and is conserved, while in another frame $E'\neq H'$, and $H'$ is conserved while $E'$ is not? The following happened playing Minesweeper:   How do I figure out which square to click? Proving inequality $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3\sqrt[3]{abc}}{a+b+c} \geq 4$ I started to study inequalities - I try to solve a lot of inequlites and read interesting .solutions . I have a good pdf, you can view from  . The inequality which I tried to solve and I didn't manage to find a solution can be found in that pdf but I will write here to be more explicitly.  Exercise 1.3.4(a) Let $a,b,c$ be positive real numbers. Prove that $$\displaystyle \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3\sqrt[3]{abc}}{a+b+c} \geq 4.$$ (b) For real numbers $a,b,c \gt0$ and $n \leq3$ prove that: $$\displaystyle \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+n\left(\frac{3\sqrt[3]{abc}}{a+b+c} \right)\geq 3+n.$$