I'm given the task of calculating the sum $\sum_{i=0}^{n}\sin(i\theta)$.  So far, I've tried converting each $\sin(i\theta)$ in the sum into its taylor series form to get: $\sin(\theta)=\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\frac{\theta^7}{7!}...$ $\sin(2\theta)=2\theta-\frac{(2\theta)^3}{3!}+\frac{(2\theta)^5}{5!}-\frac{(2\theta)^7}{7!}...$ $\sin(3\theta)=3\theta-\frac{(3\theta)^3}{3!}+\frac{(3\theta)^5}{5!}-\frac{(3\theta)^7}{7!}...$ ... $\sin(n\theta)=n\theta-\frac{(n\theta)^3}{3!}+\frac{(n\theta)^5}{5!}-\frac{(n\theta)^7}{7!}...$  Therefore the sum becomes, $\theta(1+...+n)-\frac{\theta^3}{3!}(1^3+...+n^3)+\frac{\theta^5}{5!}(1^5+...+n^5)-\frac{\theta^7}{7!}(1^7+...+n^7)...$  But it's not immediately obvious what the next step should be.  I also considered expanding each $\sin(i\theta)$ using the trigonemetry identity $\sin(A+B)$, however I don't see a general form for $\sin(i\theta)$ to work with.
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?
In the two following snippets, is the first one safe or must you do the second one?  By safe I mean is each thread guaranteed to call the method on the Foo from the same loop iteration in which the thread was created?  Or must you copy the reference to a new variable "local" to each iteration of the loop?  var threads = new List&lt;Thread&gt;(); foreach (Foo f in ListOfFoo) {           Thread thread = new Thread(() =&gt; f.DoSomething());     threads.Add(thread);     thread.Start(); }   -  var threads = new List&lt;Thread&gt;(); foreach (Foo f in ListOfFoo) {           Foo f2 = f;     Thread thread = new Thread(() =&gt; f2.DoSomething());     threads.Add(thread);     thread.Start(); }   Update: As pointed out in Jon Skeet's answer, this doesn't have anything specifically to do with threading. How do I do a case-insensitive string comparison? Recently I commented on  question   and the text did got bold but the italic word didn't seem to get italic on Android Chrome(Latest Ver.), but when I saw from the PC browser, it rendered as it should...  I've attached a screenshot of my phone ... In music, when two or more pitches are played together at the same time, they form a chord. If each pitch has a corresponding wave frequency (a pure, or fundamental, tone), the pitches played together make a , which is obtained by simple addition. This wave is no longer a pure sinusoidal wave.  For example, when you play a low note and a high note on a piano, the resulting sound has a wave that is the mathematical sum of the waves of each note. The same is true for light: when you shine a 500nm wavelength (green light) and a 700nm wavelength (red light) at the same spot on a white surface, the reflection will be a superposition waveform that is the sum of green and red.  My question is about our perception of these combinations. When we hear a chord on a piano, we’re able to discern the pitches that comprise that chord. We’re able to “pick out” that there are two (or three, etc) notes in the chord, and some of us who are musically inclined are even able to sing back each note, and even name it. It could be said that we’re able to decompose a Fourier Series of sound.  But it seems we cannot do this with light. When you shine green and red light together, the reflection appears to be yellow, a “pure hue” of 600nm, rather than an overlay of red and green. We can’t “pick out” the individual colors that were combined. Why is this?  Why can’t we see two hues of light in the same way we’re able to hear two pitches of sound? Is this a characteristic of human psychology? Animal physiology? Or is this due to a fundamental characteristic of electromagnetism? For an affine connection $\nabla$, prove the curvature R  $R(X,Y,Z,\alpha)=\alpha(\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z -\nabla_{[X,Y]}Z)$  with $X,Y,Z$ vector fields and $\alpha$ a co-vector, is a tensor.  So I realise the aim is probably to show that each of the three terms in the expression for $R$ are tensors themselves as then the result will obviously follow. However, I'm not too sure how to show each of these terms are tensors. Do I need to expand out each covariant derivate to get a bunch of connection coefficients and then see some cancellations? Still though, I'm not too sure what I'll want to see after all of this so that I can say "…therefore $R$ is a tensor".  Also, very much related to the question, I'm a bit confused with the difference between  $\nabla_Y Z$ $\space \space $ and $\space \space $  $\nabla_{\mu}\omega$.  The second one I know as the covariant derivative of a 1-form, but what is meant by the covariant derivate of a vector $\textit{field}$? As in, what's the difference between the $\nabla_Y$ and $\nabla_{\mu}$?  Edit: In answering this last question, I have realised that $\nabla_YZ=Y^{\mu}\nabla_{\mu}Z^{\nu}$ and $\nabla_{\mu}\omega=\nabla_{\mu}\omega_{\nu}$. Should you declare methods using overloads or optional parameters in C# 4.0? I was watching , and it got me thinking about when optional parameters are available in C# what is going to be the recommended way to declare methods that do not need all parameters specified?  For example something like the FileStream class has about fifteen different constructors which can be divided into logical 'families' e.g. the ones below from a string, the ones from an IntPtr and the ones from a SafeFileHandle.  FileStream(string,FileMode); FileStream(string,FileMode,FileAccess); FileStream(string,FileMode,FileAccess,FileShare); FileStream(string,FileMode,FileAccess,FileShare,int); FileStream(string,FileMode,FileAccess,FileShare,int,bool);   It seems to me that this type of pattern could be simplified by having three constructors instead, and using optional parameters for the ones that can be defaulted, which would make the different families of constructors more distinct [note: I know this change will not be made in the BCL, I'm talking hypothetically for this type of situation].  What do you think? From C# 4.0 will it make more sense to make closely related groups of constructors and methods a single method with optional parameters, or is there a good reason to stick with the traditional many-overload mechanism? Add anchors to the how-to-ask and how-to-answer pages Less than or equal sign If I know for two numbers a and b that    $${a &lt; b }$$  Then is it correct to say that    $$ a \leq b $$  I know that the second statement is true as long as the first one is. It seems OK as it is true but from the other side it seems kinda weird (to me), to say that, if you know, that a is strictly less that b.It's like you lose some extra information. Implementing support for nested transactions using JDBC 3.0 our legacy application uses JDBC 3.0. it supports transactions by implementing its own transaction manager that is capable of returning the same JDBC connection for each thread. The problem I recently discovered is that it doesn't support nested transactions: if a transaction is started inside an another transaction, then every SQLs running in the context of the inner transaction will be executed using the same db connection, and when it's commited or rolled back it will automatically commit or roll back all the changes starting from the outer transaction.  As far as I understand I can implement the support for nested transaction using JDBC 3.0 savepoints: whenever a nested transaction is started, I can set a new savepoint for the current connection. Afterward, if the nested transaction is rolled back, I will just rollback to this savepoint. If, on the other hand, it is commited, I will just do nothing. Only the commit of the most outer transaction will save the changes to the db.  Is this correct? Does this approach have any flaws? If yes, what are my possibilities?  Thanks. Where are static variables stored in C and C++? How can the contact point of rolling body have zero velocity? They say that for a rolling body, the velocity of the contact point is zero. I'm not getting this. How can it be zero when it's in continuous motion? using gb4e keeping interlinear glosses together on the same page I am using gb4e for several hundred interlinear glosses in my dissertation. Several of the examples span more than one page, that is, the vernacular and gloss will be on one page, while the free translation appears at the top of the next page.   How can I keep all three elements together on the same page? I often have subexamples (a, b, c) that don't need to stay together. That is, 1a can be on one page, and 1b on the next, as long as vernacular-gloss-free translation for each subpart stay together.  In the MWE below, the free translation is at the top of page 2, while the vernacular and gloss are at the bottom of page 1.  \documentclass[11pt,letterpaper]{article} \usepackage{lipsum} \usepackage{gb4e} \noautomath \begin{document} \lipsum[1-4] \begin{exe}     \ex \gll wuz x̃ũch it-ak xohish-em\\              1\textsc{sg.prn} bread eat-\textsc{inf} want.\textsc{prs}-1\textsc{sg}\\         \glt `I want to eat bread.'  \end{exe} \end{document}   A solution was proposed to a similar question (), but it requires me to manually look for what could be dozens of orphans throughout my dissertation (and doesn't work when applied to my example, since it still strands the FT). I would prefer a more elegant solution that takes care of the problem automatically. Looking for succinct way of describing the act of solidifying an idea into reality Looking for a succinct way of describing the act of solidifying an idea into reality. "Building [idea] out" seems too generic but I'm trying to avoid a long winded, step by step explanation of the process.  Any ideas would be greatly appreciated. Thanks! how to crop image in to pieces programmatically How to display alt text for an image in chrome The responsiveness is really bad on a newly installed Ubuntu 14.04 LTS.  Here's the specs of my laptop:  Quadcore Intel i7-4600U CPU @ 2.10GHz 12GB Ram 1TB Samsung EVO SSD   And when gradle is compiling, my entire system slows to a crawl, videos begin stuttering, browser scrolling lags... etc.  The same goes when a page refresh occurs in another window in Chrome, and while it refreshes, the HTML5 video in another tab begins stuttering.  How can I fix this?  Here are some logs:  $ vmstat -a 1 5 -   $ iostat -x 1 5 -   $ top -b - Reducibility of polynomials modulo p How do I check the Database type in a Rails Migration? What is the probability of a biased coin coming up heads given that a liar is claiming that the coin came up heads? A biased coin is tossed.   Probability of Head - $\frac{1}{8}$  Probability of Tail - $\frac{7}{8}$  A liar watches the coin toss. Probability of his lying is $\frac{3}{4}$ and telling the truth is $\frac{1}{4}$. He says that that the outcome is Head. What is the probability that the coin has truly turned Head?  My Attempt:  I used the formula: $$P(A \mid B) = \frac{P(A\cap B)}{P(B)}$$  $\ \ \ \ \ \ $P(it is head GIVEN liar said it's head) = P(it's head AND liar said it's head) / P(liar said it's head)  or, P(it is head GIVEN liar said it's head) = $\frac{ \frac{1}{8} \frac{1}{4} }{ \frac{7}{8}\frac{3}{4} + \frac{1}{8}\frac{1}{4} }$ [using a probability tree will be helpful here]  or, P(it is head GIVEN liar said it's head) = $\frac{1}{22}$  The Question: Is the method I used wrong in any way? Some others I have talked to are saying the answer will be $\frac{1}{4}$. Their reasoning is this: since the liar lies 3 times out of 4 and he said it is head, then the probability of it being head is 1/4. So who is right? What will be the answer? What is the limit of series $\frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{4 \cdot 5 \cdot 6} + \frac{1}{6 \cdot 7 \cdot 8} + \cdots $?  The $n$th summand is $\frac{1}{(2n)(2n + 1)(2n+2)} = \frac{1}{4} \frac{1}{n(2n+1)(n+1)}$.  I have tried expressing this as a telescoping sum, or as the limit of Riemann sums of a partition (the usual methods I normally try when doing this type of question- what are some other strategies?)