The computation of Fourier series coefficients.
Fourier Series for $|\cos(x)|$
How to justify the definition of summation $s_n=\sum_{i=1}^n a_i=a_1+\cdots+a_n$? I read  and found that they define summation via recursion, so I decided to formalize the proof that that this definition is actually valid. I've two questions:   Does my proof contain any error? Are there other simple ways to define summation?   Thank you so much!     Suppose that $(a_1,\cdots,a_n)$ is a finite sequence in $\mathbb N$. Show that there is a sequence $(s_1,\cdots,s_n)$ such that $s_1=a_1$ and $s_{i+1}=s_i+a_{i+1}$ for all $1\leq i&lt;n$.   My attempt:  We define mapping $f$ as follows: $f: \mathbb N\times\mathbb N\to\mathbb N\times\mathbb N: (i,a)\mapsto\begin{cases} (i+1,a+a_{i+1})&amp;\text{if }i&lt;n\\ (i+1,a)&amp;\text{if }i\geq n \end{cases}$  By recursion theorem, there is a unique sequence $(p_i\mid i\in\mathbb N)$ such that $p_0=(1,a_1)$ and $p_{i+1}=f(p_i)$. Let $\pi:\mathbb N\times\mathbb N\to\mathbb N$ be the projection to the second co-ordinate i.e. $\pi(i,a)=a$. Let $s_i=\pi(p_i)$ for all $1\leq i\leq n$, then $(s_i\mid 1\leq i\leq n)$ is the required sequence. It's clear from the definition of $s_i$ that $s_{i+1}=s_i+a_{i+1}$ for all $1\leq i&lt;n$. If $|G| = pq$,  $p,q$ primes, $p \gt q, q \nmid p-1 $, then how do I prove $G$ is cyclic without using Sylow's theorems? What would be the effects of warp collisions? Triggered by a  I was wondering what actually would happen to a ship (e.g. the Enterprise-D NCC-1701) that collides with a stationary object such as a) another ship of a similar configuration or b) a gigantic structure like a moon or even a planet.  We know the deflector is supposed to keep interstellar dust (not to be confused with ordinary dust, which is a different thing) from ripping large holes in the ship.  At first, one thinks "Obvious. ." But does it really? Would the deflector manage to actually deflect both masses? If not, where is the threshold of what a deflector can handle? Would the ship maybe tunnel right through the object, because of wibbly-wobbly space wobblification by the warp field? It does do some weird things to the space right in front and behind the vessel.  Important remark: We cannot approach this with what we know about physics at all. For all intents and purposes, both ships must measure the other ship travelling faster than light, resulting in it having infinite (or rather undefined) mass and by extension momentum. If we start with that reasoning any ship jumping to warp would suck in the entire universe (it takes a moment, but think about it). So, obviously we cannot apply our intuition (even our relativistic intuition) here. Coconut Water from White vs Brown Coconuts Class moderncv moving footer I am using documentclass moderncv to style my resume, and I'd like to put the information that normally goes at the bottom of the document, such as phone number and email, to the space next to the name aligned on the left of the document. I'm a beginnner user, so I'd love to get some enlightenment on this.  \documentclass[11pt,letterpaper,sans]{moderncv}  % moderncv themes \moderncvstyle{casual} \moderncvcolor{blue} \usepackage[scale=0.75]{geometry}  % personal data \firstname{John} \familyname{Doe} \mobile{+1~(234)~567~890}  % content  \begin{document} \makecvtitle \pagestyle{empty}  \section{Education} \cventry{1998-2007}{Degree}{Institution}{City}{\textit{Grade}}{Description}   \section{Computer skills}  \thispagestyle{plain} \end{document}    Simply put, I want to move \mobile from the bottom to the top left. Is it possible to use a laptop as an external monitor? I need to create a multi-monitor setup for my desktop computer but I have no additional monitors with me right now, aside from my laptop. Is it possible to use my laptop screen as an external monitor?  Note that I am not trying to connect a monitor to my laptop, rather, I am trying to connect my laptop screen to a desktop video card (if this is possible). In Fallout 4 the Bobbleheads are back.  Once you pick them up they stay in your inventory with a weight of 0 and a value of 300. Will I need them later ? Do I lose the bonus if I sell them or drop them somewhere ? Same with the magazines.  Is there any disadvantage to selling them ? I've heard that you always want the same size of RAM (2x1GB sticks) but does their placement matter? I have four slots on my motherboard but does it matter if the sticks are not adjacent?  This particular question involves the ArrayList contains case sensitivity Pumpkin Noodles is a thing or are a thing? Which of the following is correct?  How come pumpkin noodles is not a thing?  or  How come pumpkin noodles are not a thing?  I want to post one of the above statements to Twitter with a picture of a bowl of pumpkin noodles i.e. the stuff that comes out of a hollowed out pumpkin. How would I be able to create a grill of sorts found on the front of some computer cases such as this one I might end up teaching philosophy in Finnish lukio/gymnasium. I will not get an opportunity to practice teaching it - I do get to practice teaching mathematics a fair deal, though. (The situation is somewhat unfortunate.) I have taken subject studies in philosophy at university. They amount to around 60 credits (ECTS).  Suppose I give the students an exam, which asks them to write an essay or two. After that my task is to grade the exams in a fair way and comment them in such a way that the students can learn from the feedback.  -  Suppose the task was to write an essay on the problem of universals - nominalism and realism, say. Let us say 6 points is the perfect score for this task.  I would expect the students to define nominalism and realism (2 points, maybe?) and explain the problem (2 points). Remaining two points for taking some stance and arguing for it - realism is more reasonable because so and so. Partial credits for demonstrating only partial understanding.  -  Am I missing some important aspects of philosophical thinking? Should I emphasize the different parts of an answer in a different way? I suspect my perspective is influenced by fairly extensive mathematics education - is there something particular I should watch out for? There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$? There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$, what I tried is if possible then define $g(z)=f(z)-{1\over 2^{1\over z}}$ Then $g({1\over n})=0$ and so $g(z)$ is entire and its $0$ set has limit point in it and so $f(z)={1\over 2^{1\over z}}$ which is not analytic at $0$? Please help!  Edit: OOps! the way I defined $g(z)$ that is not entire! could any one give me hint? Why doesn't backpropagation work when you initialize all the weight the same value (say 0.5), but works fine when given random numbers?   Shouldn't the algorithm calculate the error and work from there, despite the fact that the weights are initially the same? I have to determine how many six-integer subsets of $\{1, 2, 3, \ldots, 20\}$ are possible if no two consecutive integers are in a set. Regular (non-moderator) users should be able to override deletions by Community user There is a question on  that got deleted by the Community user. This question isn't terrible (we have worse one's there), and I'm not complaining about this question in particular.  I wanted to vote to undelete it. However, it says "A moderator has deleted this post and it cannot be undeleted". This "moderator" is Community!  I understand if a "real" live human moderator (no offense to all non-humans out here) deleted this question that we can't override it. But this is a Robot! It sees that it got a downvote, no answer and little activity for a while and just deletes it.  Can we have "Community user" have rights to delete, but have those "deletions" subject to normal-person override? Proving $\sum\limits_{i=1}^k | \langle x,v_i \rangle \langle y,v_i\rangle| \leq \|x\|\cdot \|y\|$ Let $V$ be a real inner product space, and let $v_1,v_2, \dots ,v_k$ be an orthonormal set of vectors. How do you prove that  $$\sum_{i=1}^k | \langle x,v_i \rangle \langle y,v_i\rangle| \leq \|x\|\cdot\|y\|?$$  When does the equality hold?  I've been trying to do this with the Bessel and Cauchy-Schwarz inequalities, but I can't make it work yet. Any help would be greatly appreciated. What packages do people load by default in LaTeX? I'm getting the impression from reading the answers written by some of the real experts here that there are quite a few little packages that just tweak LaTeX2e's default behaviour a little to make it more sensible here and there.  Rather than try to pick these up one by one as I read answers to questions (and thus risk missing them), I thought I'd ask up front what LaTeX2e packages people load by default in (almost) every document.  As this is a "big list" question, I'm making it CW.  I don't know if there are standard rules across all SE/SO sites for such questions, but on MathOverflow the rule is generally: one thing (in this case, package) per answer.  I guess that if a couple of packages really do go together then it would be fine to group them.  This is perhaps a little subjective and a little close to the line, so I'll not be offended if it gets closed or voted down!  (But please explain why in the comments.)  Also see our community poll question: Product of two cyclic groups is cyclic iff their orders are co-prime Say you have two groups $G = \langle g \rangle$ with order $n$ and $H = \langle h \rangle$ with order $m$. Then the product $G \times H$ is a cyclic group if and only if $\gcd(n,m)=1$.  I can't seem to figure out how to start proving this. I have tried with some examples, where I pick $(g,h)$ as a candidate generator of $G \times H$. I see that what we want is for the cycles of $g$ and $h$, as we take powers of $(g,h)$, to interleave such that we do not get $(1,1)$ until the $(mn)$-th power. However, I am having a hard time formalizing this and relating it to the greatest common divisor.  Any hints are much appreciated! Which of those functions are PRFs?