I am having hard time deciding whether $\sum_{n=1}^{\infty} \frac{\sin(n)}{n}$ converges or diverges.    Intuitively I want to say that it converges, because it is like the alternating harmonic series. None of the tests such as the limit comparison test or root tests are applicable on this, so can someone help me out?
I am a beginner with Fourier series and I have to evaluate the sum   $$\sum_{n  =1}^{\infty}{\sin\left(n\right) \over n}$$  I don't know which function I have to take to evaluate the fourier series ... Someone can give me a hint ?  Thanks in advance!
I always thought water sources were called taps in kitchens, bathrooms etc, but a Google search only returned outdoor taps.  So my question is, what are indoor taps really called? Like this one: How to reconcile the cashless human/Federation society with human traders and merchants From a variety of sources, including , we know that the Federation (or at least the humans) don't use money. This raises the problem that humans frequently visit worlds where there is currency, and, in the case of DS9, operate a station that has currency frequently in use (for example, Federation personnel frequently attend Quark's bar). Additionally, we see human traders.   My question is, how do humans get "money" (credits, latinum, etc.)? How would a human even get the resources to purchase their own merchant ship or the resources to fill its cargo holds with merchandise? If Federation personnel can simply be given money for the asking, why haven't they poured so much money down Quark's throat that he chokes on it? What should a rational person accept as a miracle? How would I prove there is no smallest positive rational number? what is the best method to prove this statement? Does a conducting wire give off measurable radiation? Using FFmpeg and IPython Does existence of $\gcd$ implies PID? We know that if $A$ is a PID, then we can guarantee the existence of a $\gcd$ between two elements. My question is, does the converse hold? If I know that every two elements have a $\gcd$, do I know that the ring is a PID? My intuition says no, in that case I would like a counter-example.  Thanks. Does and has both are used with singular pronouns (He has the bottle , He does play cricket , etc) whereas Do and have are used with plural pronouns ( They have the bottle , Do they like cricket? , etc)  But still we use Does with have ( She does have a car ). Why? Shouldn't it be     She does has a car. I think the title says it all. I can't see any reason employers have to incentivize employees to contribute to their 401(k) evenly throughout the year. Instead couldn't they use a relatively simple formula that is based on how much of their income the employee has contributed to their 401(k) thus far for the year? If that or their current contribution is over the threshold, they get the match this pay period, even if they didn't contribute anything.  EDIT: To clarify, my employer matches 100% of the first 3% of my contribution and 50% of the next 2%. The match is added every pay period along with my contribution. I am pretty sure there is no "true-up" feature, so if I were to hit the $17.5K limit early, and not be able to contribute at least 5% of my salary, I would not get the full match. So I have to be careful to make my contributions evenly throughout the year to get the maximum match. Just trying to figure out my employer's motivation for requiring this. Edit Mail rules manually When I reinstall Windows XP by mistake I deleted all five partitions. I did not format them yet though.  How can I recover the partitions? When I attach the hard disk to another PC it just says that the partition is unformated and asks "Would you like to format it?". Indefinite articles used with plural nouns: It was AN amazing TWO DAYS After deleting all nodes, how do I reset the node ID to start from 1 again? I generated tons of dummy content for testing, and then deleted them all.  As nid is auto-increment, newly created node will have a nid of last created nid +1.  How can I make nid start from 1 again? Pullback of locally free sheaves is locally free states that if $f:X \rightarrow Y$ is a morphism of ringed space, $G$ is a locally free $O_Y$-module, then $f^*G$ is a locally free $O_X$ module.     Suppose that $G$ is a locally free $O_Y$ module, hence it is a free $O_Y|_U=O_U$ module. We have an induced map  $$f^{-1}(U) \rightarrow U$$ given by restriction of morphism of ringed spaces.   Hence, as $\bigoplus O_U \simeq G|_U$, $f^*:Mod(O_U) \rightarrow Mod(O_{f^{-1}(U)})$ is functorial, $f^*(G|_U) = (f^*G)|_{f^{-1}(U)}$, we have isomoprhism,  $$ \bigoplus O_{f^{-1}(U)} \simeq (f^*G)|_{f^{-1}(U)} $$ I was on the tube and overheard a dad questioning his kids about maths.  The children were probably about 11 or 12 years old.   After several more mundane questions he asked his daughter what $1/0$ evaluated to. She stated that it had no answer. He asked who told her that and she said her teacher. He then stated that her teacher had "taught it wrong" and it was actually $∞$.  I thought the Dad's statement was a little irresponsible.  Does that seem like reasonable attitude? I suppose this question is partly about morality. Do paths in Adobe Photoshop have to be closed? I am working to convert a png to a vector graphic.  I am outlining the image using the pen tool.  Photoshop insists that the pen tool produce closed loops in order for me to use the "Stroke path..." tool.  Why is this? What is Stack Overflow's business model? I once , but that was . Now, , and they have , but where's the business model? The  its own Q&amp;A about its site (since it wasn't included in the ), and just about the only dig in there that halfway has merit is the business model dig:  We switched to a premium model to keep out of the Venture Capital business (been there, done that, got the t-shirt). Companies like StackExchange couldn't do it without the VC cookie jar. Where's the model? Huh, Spolsky? (I’m sorry. Did that slip out?)  It hurt, but I can't help but wonder if they're on to something.  How would Stack Overflow Internet Services pay back the VCs? Ad revenue alone?  There is only one 'paid' product, Stack Overflow careers, but it doesn't seem like that could sustain a business as large as Stack Overflow (Joel once mentioned on his blog that his jobs board brought in an extra million a year in revenue). So what does Stack Overflow do when the VC cash runs out? Is this a get-big-and-take-the-company-public maneuver? Is that the only viable business model for Stack Exchange? Edit: With the size of Stack Overflow's team, it isn't a stretch to say that payroll is at least $2 million  a year, probably as much as 4 when you think about benefits (average 120K for a developer in New York City; x2 for benefits/taxes/all-that-other-good-stuff). Office space can't be cheap, so how long does $6 million last before you need another VC infusion? Microphone Not Working With FaceTime How to install the Boot-Repair tool in an Ubuntu live disc? When the bootloader (GRUB) is broken, I cannot access neither Ubuntu nor Windows.  I heard that an easy solution to fix this is to use the Boot-Repair tool from an Ubuntu liveCD (or liveUSB).  What are the steps to do this? Low Quality Posts reviews add comments more than once on behalf of reviewer