I need to prove that   $\forall n\in\mathbb N:$  $$\sum_{i=1}^n \sin(\frac{i\pi}{3}) = 2\sin(\frac{n\pi}{6})\sin(\frac{(n+1)\pi}{6})$$  Using induction only leads me to proving quite complicated trig identities. I guess it also can be solved using reminders because sins on the LHS are equal to:  $$\frac{\sqrt3}{2},\frac{\sqrt3}{2},0,-\frac{\sqrt3}{2},-\frac{\sqrt3}{2},0$$ so per every 6th term they cancel out.  Could You give me any hint?
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?
I have the below list of files  aro_tty-mIF-45875564pmo_opt aro_tty-mIF-45875664pmo_opt aro_tty-mIF-45875964pmo_opt aro_tty-mIF-45875514pmo_opt aro_tty-mIF-45875524pmo_opt   that I need to rename to  aro_tty-mImpFRA-45875564pmo_opt aro_tty-mImpFRA-45875664pmo_opt aro_tty-mImpFRA-45875964pmo_opt aro_tty-mImpFRA-45875514pmo_opt aro_tty-mImpFRA-45875524pmo_opt traveller's cheques; customer's requirements: use of the singular Saxon genitive (and the plural attributive) for association May I ask if the following sentence is grammatical?      Our remit is to conduct research on customer's requirements.   In my understanding, as 'customer' is a countable noun, so the plural form 'customers''' should be used instead for this general reference (without specification of certain customers).   However, I know that we can say 'traveller's cheques' - e.g.,      Traveller's cheques are available at the bank.    In this sense, actually can we say 'customer's requirements' in the first case? How can I convert a Unix timestamp to DateTime and vice versa? There is this example code, but then it starts talking about millisecond / nanosecond problems.  The same question is on MSDN, .  This is what I've got so far:  public Double CreatedEpoch {   get   {     DateTime epoch = new DateTime(1970, 1, 1, 0, 0, 0, 0).ToLocalTime();     TimeSpan span = (this.Created.ToLocalTime() - epoch);     return span.TotalSeconds;   }   set   {     DateTime epoch = new DateTime(1970, 1, 1, 0, 0, 0, 0).ToLocalTime();     this.Created = epoch.AddSeconds(value);   } } Why do PBX systems use -48 V? It seems common that PBX and other telephone hardware use a positive-ground power supply, where the "hot" line is at -48v.  What's the reason for that? Suppose I'm given the mean and one quantile (e.g. the 20% quantile) of a random variable $x$, and I want to find the parameters $\alpha$ and $\beta$ of a Beta distribution that has the same mean and quantile.  Is there an efficient way to do it?    Using just the mean, I know that since $\bar{x} = \frac{\alpha}{\alpha+\beta}$, we have $\beta = \frac{\alpha}{\bar{x}} - \alpha$.  So we only really have one parameter to estimate.  But I'm unsure how to use the quantile information to take the next step.  Maybe there's something I can do with the Incomplete Beta when I know the ratio $\frac{\beta}{\alpha} = \frac{1-\bar{x}}{\bar{x}}$?  I have access to R myself, so I could use a numerical optimizer for this, but ideally I need a method that can be carried out in Excel in someone else's environment.  Excel does have BETA.DIST() and BETA.INV() functions available.  A look-up table would be fine, but a closed-form formula would be better if it's possible. Why can't I destroy or place blocks? Movie Identification - 1990s VHS Sci-Fi/Horror To find:  $$\int \sqrt{1+\sin(x)} dx$$  What I  tried:  I put $\tan(\frac{x}{2}) = t$, using which I got it to:  $$I = 2\int \dfrac{1+t}{(1+t^2)^{\frac{3}{2}}}dt$$  Now I am badly stuck. There seems no way to approach this one. Please give a hint. Also, can we initially to some manipulations on the original integral to make it easy? Thank you. "Specially" vs "especially" When should I use specially and when especially? How can I tell if sheet a exists in a workbook? I have the following code that's not working as expected:  If Sheets("a") &lt;&gt; "" Then MsgBox ("sheet a exists")   How can I tell if sheet a exists in the workbook? Use the view from other cameras in the scene as VideoTexture in Cycles I have a scene with monitors that display video from cameras in the same scene (or other scenes, but all of them are linked i.e., same objects in the scene). The cameras shown on the monitors do not view the monitors, so no looping should occur.  I found  that shows how to get the effect in the Game Engine.  I would like to do this in cycles.  Essentially I would like a material input node that corresponds to the output of the composition of a particular scene. The incoming image should honor the render settings of the particular scene and any compositing applied to it.  The Render Layer node does not appear to honor the resolution or compositing settings of the scene it comes from, and is not available to apply as a texture in the material node editor (that I can find).  I understand that this could be done by rendering the animation from the various cameras, then applying that video as textures to the monitors, then rendering the camera of the monitors. That's super tedious if your making changes to the whole scene, and I don't want to do that. Postfix Connection Timed out I used the  tutorial below to setup postfix and dovecot on a Digital Ocean server. But I'm getting "Connection Timed out" when postfix/smtp tries to deliver an email to a remote host like gmail or yahoo..      RECEIVING A NEW MESSAGE /var/log/mail.log    Feb 2 22:33:38 localhost dovecot: auth-worker: Debug: Loading modules from directory: /usr/lib/dovecot/modules/auth  Feb 2 22:33:38 localhost dovecot: auth-worker: Debug: pam(app,189.63.49.XXX): lookup service=dovecot  Feb 2 22:33:38 localhost dovecot: auth-worker: Debug: pam(app,189.63.49.XXX): #1/1 style=1 msg=Password:   Feb 2 22:33:38 localhost dovecot: auth: Debug: client out: OK#0111#011user=app  Feb 2 22:33:40 localhost postfix/submission/smtpd[1045]: E9AA724264: client=unknown[189.63.49.XXX], sasl_method=PLAIN, sasl_username=app  Feb 2 22:33:43 localhost postfix/cleanup[1052]: E9AA724264: message-id=  Feb 2 22:33:43 localhost postfix/qmgr[32661]: E9AA724264: from=, size=485, nrcpt=1 (queue active)  Feb 2 22:33:45 localhost postfix/submission/smtpd[1045]: disconnect from unknown[189.63.49.XXX]  Feb 2 22:34:14 localhost postfix/smtp[1053]: connect to mta5.am0.yahoodns.net[98.136.216.25]:25: Connection timed out    Feb 2 22:24:21 localhost postfix/smtp[1013]: connect to mta6.am0.yahoodns.net[98.136.217.202]:25: Connection timed out  Feb 2 22:24:21 localhost postfix/smtp[1014]: connect to gmail-smtp-in.l.google.com[173.194.76.27]:25: Connection timed out  Feb 2 22:24:21 localhost postfix/smtp[1015]: connect to gmail-smtp-in.l.google.com[173.194.76.27]:25: Connection timed out  Feb 2 22:24:21 localhost postfix/smtp[1012]: connect to aspmx.l.google.com[173.194.68.27]:25: Connection timed out    Here is /etc/postfix/main.cf   /etc/postfix/master.cf   Result from #> netstat -nlptu Infinite product with sign changes. For infinite products I know the following theorems:  Theorem I. If $a_n \geq 0$ for all $n$ then $\prod (1+a_n)$ converges if and only if $\sum a_n$ converges.  Theorem II. If $-1 &lt; a_n \leq 0$ for all $n$ then $\prod (1+a_n)$ converges if and only if $\sum a_n$ converges.  My question is what are necessary and sufficient conditions (possibly including more restrictions than in I and II) for $\prod (1+a_n)$ convergence/divergence as related to  $\sum a_n$ convergence/divergence  if the sequence of terms $a_n$ can switch signs repeatedly but always stay greater than $-1$.  I already know the trivial equivalence of $\prod(1+a_n)$ and $\sum\ln(1+a_n)$. Find the summation of given expression: $\sum_{k=1}^n k^3\binom nk$ I am trying to solve the following question which is Ex3 from Arthur Engel, Problem Solving strategies. Here is the question: $\sum_{k=1}^n k^3 {n \choose k}$ and asks to find the sum. I am sincerely having a hard time trying to figure out how to solve it and get $n^2 (n+3)(2)^{n-3}$. I would really appreciate if someone could show the step-by-step process. Thank you! Bootstrap CSS Active Navigation What is safe to exclude for a full system backup? Wrapping a mesh around point cloud with cavities I have a 32-bit machine, can I use programs that are meant for 64-bit processors? I am using a 32-bit machine, but I am having to use a relatively obscure semantic database which only comes in 64-bit versions.  What are my options for running it in my local dev environment?  Currently when I try to run it, I get an error message that looks like this:  cannot execute binary file   and when I check my machine info, I get this:  $ uname -a Linux slabrams-desktop 2.6.32-29-generic #58-Ubuntu SMP Fri Feb 11 19:00:09 UTC 2011 i686 GNU/Linux triggered my curiosity for this question. Why most of manga are black and white? Is this because of more cost if they add colors to it? i have read most of the related postings and answers allready ... (e.g.  and  ... among others)  but (!) unfortunately none of them solved my problem. as someone postet on 2012-01-04 12:27 definition of properly defined *.csvt doesn't effect field-import - they are expected to be numbers and still are string.  her is a sample of my csv  InputID,Mean,Stdev,min,max 1,334.7617168,36.87450474,294.4334857,383.5583136 2,446.3946568,207.949905,153.2494524,613.3182137 and the csvt "Integer","Real","Real","Real","Real"  .. i have already:    checked field names (no blanks), checked separators, ... which are qgis' default because the csv is distance matrix calculation result defined the csvt with simle "Integer","Real" and the precision-type-style "Integer(6)","Real(8.3)" changed country settings in win7, win8, osx 10.9 due to known decimal separator issues   used csv as specified as well as "excel-style" csv (";"-separated)    checked all that in qgis 2.0, 2.2, 2.3(master)   and: at the beginning this problem was related to distance matrix output  ... where qgis calculates the results (and writes the csv-outputs), which of course are numbers (mean, stdev, min, max)  so, initially i was just looking for a straight forward way to use qgis' own calculation results for mapping of distance-distribution ...  is there a way to solve or completely bypass the csv-t issue in a way, i can easily communicate to gis-beginners ?