How would I find the sum of an infinite number of fractions, where there are Fibonacci numbers as the numerators (increasing by one term each time) and numbers (starting at one) which double each time as the denominators?  I'm assuming this has something to do with limits.
I presented the following problem to some of my students recently (from Senior Mathematical Challenge- edited by Gardiner)  In the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, 34, 55,\ldots$ each term after the first two is the sum of the two previous terms. What is the sum to infinity of the series: $$\frac{1}{2} + \frac{1}{4}+ \frac{2}{8} + \frac{3}{16} + \frac{5}{32} +\frac{8}{64} + \frac{13}{128} +\frac{21}{256} +\frac{34}{512}+ \frac{55}{1024} + \cdots$$  Now, I solved this using an infinite geometric matrix series (incorporating the matrix version of the relation $a_n= \frac{a_{n-1}}{2}+ \frac{a_{n-2}}{4}$), and my students, after much hinting on my part, googled the necessary string to stumble across Binet's formula (which allows one to split the series into two simple, if rather messy, geometrics). Both of these are good methods, but neither really seems plausible for a challenge set for 15-18 year olds under exam conditions. So how is one supposed to do it?
How to calculate the following sum: How may one calculate  $$\lim_{n\to\infty} \ \left(\left(\sum_{k=1}^{n} \frac{1}{3k-1}\right) - \frac{\ln n}{3}\right) \ ?$$ I don't have complex analysis at my beck and call, and I only have a low level of knowledge in topology, but I need to prove that this metric space (for any real $r$ and $R$ with $r &lt; R$)$$ X = \{ (x, y) \in \mathbb{R}^2 \ | \ r \leq x^2 + y^2 \leq R \}$$  with the Manhattan metric $d((x_1, y_1), (x_2, y_2)) = |x_1-x_2| + |y_1-y_2|$ is not simply connected.  I've already prooven that it's path connected, and now I need to show there are some points $P$ and $Q$ with two paths between them such that one cannot be continuously 'morphed' into the other.  $ \ $  What I have so far is as follows:  I take $P = (0, r)$ and $Q = (0, -r)$, with $f_0$ being a path from $P$ to $Q$ going clockwise around the circle radius $r$ and $f_1$ being much the same but going counterclockwise.  Now I assume there is a function $g : [0, 1]^2 \to X$ such that:   $g(s, 0) = f_0 (s)$ $g(s, 1) = f_1 (s)$ $g(0, t) = P$ $g(1, t) = Q$   To get the final result, I need to show that this function cannot be continuous, but for the life of me I cannot.  For some context, these are the topics which have been visited during the course, roughly in order of recentness.   Multiple connectedness Simple connectedness Pathwise connectedness Interior points Boundary points Open sets Compactness Complete metric spaces Bounded metric spaces Totally bounded metric spaces Closed sets Closure of a metric space Limit points Cauchy sequences Convergence Continuity Metric space quivalence Metric equivalence   Edit: I might have an argument that works, though it's far from rigourous. We can shrink $R$ to be as close to $r$ as we want, so we can essentially constrain the annulus down to a circle and thus force any path from the left side of the circle to the right side to go through $P$ or $Q$.  So holding $s \in (0, 1)$ constant and varying $t$ must produce a path through $P$ or $Q$ for any $s$. If for some $s$ it passes through $P$ and for some other $s$ it passes through $Q$, then there must be $s_0$ such that $\forall \epsilon &gt;0  \ \exists \delta \leq \epsilon$ st. $s_0$ produces a path through $P$ and $s_0 + \delta$ produces a path through $Q$.  Now we can consider the path $g(s, \frac{1}{2})$, and note that it must have a discontinuity at $s_0$.  Now consider the case where all $s$ produce paths through only one of $P$ or $Q$. WOLOG: $P$. Now, at $t = \frac{1}{2}$, $s$ arbitrarily close to $1$ are mapped away from $Q$, but $1$ is always mapped to $Q$ by definition, so $g(s, \frac{1}{2})$ has a discontinuity at $s=1$. Therefore $g(s, t)$ is not continuous.  This argument is definitely iffy to me, if no one has their own argument (using sufficiently low level concepts), then criticism on the above would be appreciated. Solving the complex equation $\sin(z) = \cos(z)$ To find the complex numbers z satisfying $\sin(z) = \cos(z)$, can I say: $$\sin(z) = \frac{(e^{iz}-e^{-iz})}{2i}=\frac{(e^{iz}+e^{-iz})}{2}$$   and solve for z? So we then reduce this to $$-e^{-iz} = e^{-iz}$$ but this doesn't look right I want to extract infromation from logs using a shell script for a particular time range . A line in logs looks like this:  [14:58:27:176][15-10-2015]: testing1-vidya-Thu Oct 15 11:49:27 IST 2015 [14:59:27:178][15-10-2015]: testing2-vidya-Thu Oct 15 11:49:27 IST 2015 [15:00:27:179][15-10-2015]: testing3-vidya-Thu Oct 15 11:49:27 IST 2015 [15:01:27:180][15-10-2015]: testing4-vidya-Thu Oct 15 11:49:27 IST 2015 [15:01:27:181][15-10-2015]: testing5-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:182][15-10-2015]: testing6-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:183][15-10-2015]: testing7-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:184][15-10-2015]: testing8-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:191][15-10-2015]: testing9-vidya-Thu Oct 15 11:49:27 IST 2015 [15:03:27:192][15-10-2015]: testing10-vidya-Thu Oct 15 11:49:27 IST 2015 [15:03:27:193][15-10-2015]: testing11-vidya-Thu Oct 15 11:49:27 IST 2015 [15:03:27:208][15-10-2015]: testing12-vidya-Thu Oct 15 11:49:27 IST 2015 [15:04:27:209][15-10-2015]: testing13-vidya-Thu Oct 15 11:49:27 IST 2015 [15:49:27:210][15-10-2015]: testing14-vidya-Thu Oct 15 11:49:27 IST 2015 [13:02:17:209][15-10-2015]: testing13-vidya-Thu Oct 15 11:49:27 IST 2015 [15:04:27:209][16-10-2015]: testing13-vidya-Fri Oct 16 11:49:27 IST 2015 [15:49:27:210][16-10-2015]: testing14-vidya-Fri Oct 16 11:49:27 IST 2015   I need to extract data in specific intervals. for example in between 14:59 to 15:03 on 15-10-2015 as shown below:   [14:59:27:178][15-10-2015]: testing2-vidya-Thu Oct 15 11:49:27 IST 2015 [15:00:27:179][15-10-2015]: testing3-vidya-Thu Oct 15 11:49:27 IST 2015 [15:01:27:180][15-10-2015]: testing4-vidya-Thu Oct 15 11:49:27 IST 2015 [15:01:27:181][15-10-2015]: testing5-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:182][15-10-2015]: testing6-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:183][15-10-2015]: testing7-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:184][15-10-2015]: testing8-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:191][15-10-2015]: testing9-vidya-Thu Oct 15 11:49:27 IST 2015 [15:03:27:192][15-10-2015]: testing10-vidya-Thu Oct 15 11:49:27 IST 2015 [15:03:27:193][15-10-2015]: testing11-vidya-Thu Oct 15 11:49:27 IST 2015 [15:03:27:208][15-10-2015]: testing12-vidya-Thu Oct 15 11:49:27 IST 2015   I am new to shell script.I have tried with grep command . Can anyone suggest me how to do this ? Seeking shapefile of city boundaries in US? What is the simplest and most robust way to get the user's current location on Android? Dobble card game - mathematical background In  there is a deck of 55 cards. Each of them has 8 symbols on it. There is a total of 50 different symbols and each two cards have one and only one in common.  I am wondering, given a total number of symbols N and a number of symbols on each card K, how do you find the biggest number of cards such that:   on each of those cards there are exactly K of the N symbols, every two cards have one and only one symbol in common. Near the end in amongst the battle, Kylo Ren lands next to his ship that Rey stole. I understand that at a stretch (or through the will of the Force) Kylo found a TIE Fighter in the wreck he was in, but to my understanding those TIEs would not have had hyperdrives, so how did he travel that far without a carrier ship? I suppose I'm guessing/assuming that he travelled on his own (Solo if you will), and not with his Star Destroyer, as the Knights who travelled with him on his Star Destroyer were already at the destination. But if the Force willed him to find the TIE Fighter, why not a shuttle with a hyperdrive? It would be iconic enough to use in the film, surely.  How did Kylo Ren travel to another system in a civil war era TIE Fighter with no hyperdrive or carrier ship? I am having a problem understanding displacement maps. I'm starting to think that I don't understand them at all.  Clarifying My Understanding  My understanding (which might be completely wrong), is that a displacement map is a gray-scale image whereby a value of 0.5 produces no displacement, >0.5 produces displacement in the positive direction, and &lt;0.5 produces displacement in the negative direction. This, however, does not seem to match my actual observations in Blender.  I am using a displacement map within Cycles, plugging it into the Displacement socket on the Material Output node (I am also using the new experimental adaptive subdivision and micro-vertex displacement). I note that a value of 0.5 actually produces a displacement in the positive direction, causing the whole mesh to jump up slightly, as well as creating the peaks and valleys that you would expect for >0.5 and &lt;0.5 values respectively.  This affects my entire understanding of displacement maps. Why is this happening? Why is a value of exactly 0.5 creating displacement? Further, how are you supposed to arrange and line up objects in your scene when, in rendered view, the displacement maps will push them out of position, even if those maps are only supposed to create displacement in specific areas.  The Problem in Practice  This is the problem I am having at the moment. I have a plane which is supposed to be a snowy field, and I am using a displacement map to give some depth to the snow. In rendered view, my plane jumps up significantly. This is what lead me to investigate my understanding of displacement maps, and to this question.  Additionally, what is the correct way to increase and decrease the strength of a displacement map? At the moment, I will run them through a Math node set to multiply, however, as you might imagine, this exaggerates the issue that I am complaining about above.  To be clear, however, I am not using a math node when experiencing the "jump up" for 0.5 values. This is not the source of my problem, but another aspect to it. Combining Shimano RS500 crankset with Stages Shimano 105 5800 I've been thinking about getting a power meter for a while now. I finally decided I would get one, but now I realise that Stages only starts at the Shimano 105 level.  I currently have a Shimano RS500 Crankset ( 50 x 34 ) and I'm very pleased with it, so I don't feel a need to replace the whole crankset with a 105 5800.  I also have no plans to upgrade anytime soon to Dura Ace or something like that (the rest of my bike has the 105 5800 set), so I don't think I should buy a higher lever crank arm from Stages.  Now, I know something about bikes, but not a lot. It seems that what I have (RS500) looks compatible with the 105 crank arm (just judging by the looks), but I would like to be certain before I spend this amount of money.  My set:  Stages I would like to get:   Kind regards and thanks in advance, What book describes consuming vials of vampire blood to pseudo-Rage? Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X Here X is a Hausdorff Space.  I know that singleton sets, {x}, are closed in a Hausdorff space. Although Im not sure if thats how to use the Hausdorff property.   Should I investigate $h=F(x)-x$?  Can anyone give a hint How can I render bokeh circles?  Nothing I do seems to work.   I tried using an empty as the focus point for a cube, with some point lamps in the background with about 2000 strength, but it doesn't work. It just makes everything blurry.  I also raised the blades to 4/5/6/12, and it doesn't make a difference.  I'm using Blender 2.74. Exact sequence of sheaves with non exact sequence of global sections Let $X$ be some topological space. By $\mathcal{F}_i$ we denote some sheaves of abelian groups on $X$. The sequence of sheaves and morphisms  $$\mathcal{F}_1\longrightarrow \mathcal{F}_2\longrightarrow \mathcal{F}_3\longrightarrow... $$ is said to be exact if for each $x\in X$ the corresponding sequence of stalks $$(\mathcal{F}_1)_x\longrightarrow (\mathcal{F}_2)_x\longrightarrow (\mathcal{F}_3)_x\longrightarrow... $$ is exact. However if the sequence of sheaves is exact than the sequence of global sections is not necessarily exact! (The most famous example is the sequence of sheaves $$0\longrightarrow\mathbb{Z}\hookrightarrow \mathcal{O}\stackrel{\exp}{\longrightarrow} \mathcal{O}^*\longrightarrow0$$ considered as sheaves on $\mathbb{C}-\{0\}$, where $\mathcal{O}$ is a sheaf of holomorphic functions, $\mathcal{O}^*$ is a sheaf of holomorphic functions with no zeros).  So, could you give me some easy examples of such phenomenon ? Why do German umlauts and sharp s (ß) work without a package in LaTeX? I am used to the fact that LaTeX will only know letters like ä, ö, ü, or ß if I use packages like  \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage[T1]{fontenc}   Today, I tried LaTeX on a Mac for the first time, using the MacTeX distribution and texmaker. Without writing down any packages, all umlauts and ß were accepted correctly from scratch. Why is that and will the usage of special letters without using the packages cause any problems? Advanced format partition alignment I cloned a non Advanced Format drive (W7 64bit) to a new hard drive (Seagate) that was AF, the software I used supposedly does not support 4k very well (Todo Backup Free 3.5).   Is there a Windows utility to check alignment? I know of the Western Digital tool but not sure if it works on Seagate brand drives. If the partitions are not aligned, Am I better to clean install W7 64bit or do the partition alignment procedure? If so, Why?   It seems to be running fine so far. From what I have read Seagate drives use  technology to help with misaligned partitions on the fly, might explain why it runs well for me, but would still like to have the partitions aligned properly.  EDIT: Here is the diskpart output for my disk partitions  DISKPART&gt; list partition    Partition ###  Type              Size     Offset   -------------  ----------------  -------  -------   Partition 1    Primary            188 MB    31 KB    Partition 2    Primary            698 GB   188 MB Story about a deadly alien creature on another planet that has some chemical in its body that makes it faster I can not remember if it was a short story or a novel, but I am leaning towards it being a novel. There is a colony of humans and I can't remember if they crash-landed or not.   There is a creature that they come to discover after it has killed a few people. The creature has some kind of chemical in its body that is like a super-adrenaline. One part is I remember is one of these creatures getting very hot from using up its reserve of this chemical and it dived into a river or pond and it started to steam. Something like that.  It has to be at least 10 years since I read the story. Is it possible to create 3D red/cyan anaglyphs in a hand drawing? There are plenty tutorials online on how to create 3D red/cyan anaglyphs from existing photos in Photoshop or similar software. However, I'm wondering if it is possible to draw a 3D image by hand on for example a whiteboard using markers.  My plan is to tape together a red and blue marker and draw with it using the same rotation/inclination of the marker across the drawing. Will this give a 3D effect when viewed through red/cyan glasses? Or is the 3D anaglyph method more complex than just offsetting red and cyan lines? Two trains travel on the same track towards each other, each going at a speed of 50 kph. They start out 100km apart. A fly starts at the front of one train and flies at 75 kph to the front of the other; when it gets there, it turns around and flies back towards the first. It continues flying back and forth till the two trains meet and it gets squashed. How far did the fly travel before it got squashed?  The question is relatively simple - the trains take an hour to collide, and in an hour the fly would travel 75 km. However, the question got me wondering: Lets say you are given a time $t$ in hours(where $t&lt; 1$) How many &quot;rounds&quot; would the fly make in $t$ time? A round  is defined as moving from one train to another and back. How would I model the equation? On the usage of "the" article with acronyms and initialisms I've read  and  I know that  1. The article is often used with initials that are pronounced letter by letter (initialisms).  2. The article is often NOT used with initials that are pronounced as a word (acronyms).  I used 'often' because I'm not sure about possible exceptions.  In my field of study (Operations Research), initials are frequently used to represent optimization problems (e.g., the TSP). I discussed this point with my university supervisor who makes comments on my reports (English is not his native language). He responded that:  "What I learned from [...] was that whenever we use (specific) acronyms, we don't use 'the' when they are within a sentence, and we use 'the' when a sentence starts with them. For instance,  (i) The HCP is an NP-Complete problem.  (ii) We developed a new algorithm to solve HCP."  HCP stands for the Hamiltonian Cycle Problem. It is really confusing me because I do not know when to use and when not to use the article. For example, on , the author Jane Watson writes:     Use a definite article with an initialism if the spelled out term begins with “the” but is not covered in the initialism.   which implies that I should use the article with "HCP". What do you think?