We have that $(x_n)$ is a sequence of real numbers. And the relation on the title: $$ |x_{n+1} - x_n| &lt; \frac{1}{3^n}. $$ We must prove that this is a Cauchy sequence. I know that an Cauchy sequence follows the definition: given $\epsilon&gt;0$, exists $n_0 &gt; 0$, such that $m,n &gt; n_o \Rightarrow |x_m - x_n|&lt; \epsilon$ But I don't know how to use both informations to prove the exercise. If someone please may help me, I'd be very thankful.
Let $(x_n)_{n=1}^{\infty}$ be a sequence such that $|x_{n+1} - x_n| &lt; r^n$ for all $n \geq 1$, for some $0 &lt; r &lt; 1$. Prove that $(x_n)_{n=1}^{\infty}$ is a Cauchy sequence.  I understand that a Cauchy sequence means that for all $\varepsilon &gt; 0$ $\exists N$ so that for $n,m \ge N$ we have $|a_m - a_n| \le \varepsilon$. But this one is really giving me a headache.  I tried doing something like: let $ m &gt; n$. Therefore $x_n - x_m$ =  $(x_n - x_{n-1}) + (x_{n-1} - x_{n-2}) + ... + (x_{m+1} - x_m) $ and then somehow using the triangle inequality to compute some sum such that $x_n - x_m$  &lt; sum which would be epsilon?  any help is appreciated, thank you.
Why avoid increment ("++") and decrement ("--") operators in JavaScript? I forgot my iPhone 5 passcode when I was changing the code. A window appeared on the device screen saying that iPhone is disabled.  I have Apple ID and password for it. After signing in to iCloud I have erased the device in order to remove the passcode from my iPhone but it still says that device will be erased when device will connect to internet. What would i have to do? Add default syntax highlighting for the [dart] tag I just approved several edits from a user adding  to questions with the  tag. Apparently , Google Code Prettify has added support for Dart.  I suggest it's time to follow suit and update the default syntax highlighting associated with the  tag. Adding it manually to every question sounds like way too much work, even if I all I have to do is click the "Approve" button. As linguistics student, I have to put more than one thing on a character every now and then. For example, I need to type ā̆. I thought I could simply do that by typing \={\u{a}}, but then the macron is placed above an empty space in front of the character.  How can I solve this? Thanks  P.S. I'm using MiKTeX 2.9, running on Windows 8; in case it might be of use. Using maxTimeMS parameter with aggregation queries on Mongo 2.6 and Pymongo 2.7.1 What are the downsides of removing ubuntu-desktop metapackage? Accessing static member through invalid pointer: guaranteed to "work"? Getting text from last marker to EOF in POSIX.2 Age replacement policy for hard disks and SSDs at servers I'm planing an age replacement policy for our storage and servers. Most of them are for DBs and some for images (static content) so yes, they have an huge I/O everytime.  Also, we use Samsung 840 Pro SSDs for the RAID Controllers (PERC H700i) as CacheCade.  Are you guys managing the replacement of old hard drives and solid state drives? How would you implement an LRU cache in Java? Please don't say EHCache or OSCache, etc. Assume for purposes of this question that I want to implement my own using just the SDK (learning by doing). Given that the cache will be used in a multithreaded environment, which datastructures would you use? I've already implemented one using  and , but I'm curious if any of the new concurrent collections would be better candidates.  UPDATE: I was just reading through  when I found this nugget:     If you need constant-time access and want to maintain the insertion order, you can't do better than a LinkedHashMap, a truly wonderful data structure. The only way it could possibly be more wonderful is if there were a concurrent version. But alas.   I was thinking almost exactly the same thing before I went with the LinkedHashMap + Collections#synchronizedMap implementation I mentioned above. Nice to know I hadn't just overlooked something.  Based on the answers so far, it sounds like my best bet for a highly concurrent LRU would be to extend  using some of the same logic that LinkedHashMap uses. What are the primary differences between the rules of D&D 3.5 and Pathfinder? What are some of the rule changes that Pathfinder brings in comparison to D&amp;D 3.5?  I am curious what changes were made so I can explain to my group what it has to offer. Override a method at instance level Socket Tester all indicators light up I used the above GFCI/RCD (up to 250v) socket tester on the 240v black and red live wire of the US split phase ac power system. When I connect the ground, all indicator lights (3 red) got on indicating "Live/Grd Reverse, Missing GRD" although it is connected properly.  What is inside the socket tester? Does it have a live and neutral leg such that it should only be used for ac system with polarity? How about if you use it on the red and black 240v phase to phase? Why would all the indicators light become on (3 red lights)?  The following is the specs of it:  Specifications: Brand Name: HABOTEST Color: Black Material: Flame Retardant ABS Plug Type: US Voltage: 48V-250V 45-65Hz RCD Test: >30mA (EU/UK Plug) RCD Working Voltage: 220V±20V GFCI Test: >5mA (US Plug) GFCI Working Voltage: 110V±20V Voltage Measurement: 48V~250V / 45~65Hz    Accuracy: ±(2.0%+2) Working Environment: 0°C ~ 40°C, 20% ~75%RH Storage Environment: -10°C ~ 50°C, 20% ~80%RH Safety Rating: EN61010-1,-2-030, EN61326-1,CAT II 300V Item Size: 62 * 65 * 55mm / 2.4 * 2.6 * 2.2in Item Weight: UK 64g/2.3oz, EU 59g/2.1oz, US 53g / 1.9oz Package Size: 7.5 * 7.4 * 7.3cm / 3.0 * 2.9 * 2.9in Package Weight: UK 90g/3.2oz, EU 86g/3.0oz, US 79g / 2.8oz Evaluate the limit: $$\mathop {\lim }\limits_{x \to \infty } x\left( {{{\left( {1 + {1 \over x}} \right)}^x} - e} \right)$$  My attempts didn't yield a result. I'd be glad for a guidance. Thanks! How does one determine ladder operators systematically? In textbooks, the ladder operators are always defined," and shown to 'raise' the state of a system, but they are never actually derived.  Does one find them simply by trial and error? Or is there a more systematic method/approach to obtain them? 15V 3A dc motor PWM frequency setting What is a suitable frequency for 15V/3A DC motor? Is there any effect on the motor when the frequency is changing? I am confused with this until now and the Internet shows a lot of theory that confuses me. Does the concept of infinity have any practical applications? I know what you're thinking: "of course it has, for example, it can be used to tell you how many times you can go around a circle". But that isn't really true, now is it? You'd be dead or the world would go under long before an infinite amount of loops had been reached.  Are there any practical applications for the concept of infinity? Is it a useful concept in maths at all?  I know that Donald E. Knuth has argued that for all practical purposes, a very, very large number has the same effect as infinity, in his book "Things a Computer Scientist Rarely Talks About" (can't remember the exact quote, nor find it online, unfortunately).  Examples are appreciated. Heroku and Twisted Ping faster than light I just discovered very strange thing while testing my internet connection. My ping is smaller than it should be. For example ping time to Arizona State University is about 14ms.  eryk@eryk-pc:~$ ping www.asu.edu PING www.asu.edu.cdn.cloudflare.net (104.16.51.14) 56(84) bytes of data. 64 bytes from 104.16.51.14: icmp_seq=1 ttl=60 time=13.8 ms   I'm living in Poznań in Poland so my distance to Arizona State University in a straight line (very optimistic assumption) is about 10000 km. Considering the fact that ping time is a time for two directions (to the target and back to home). So my ping packet have to cross 20000 km distance. Speed of light is 300 000 km per second that is 300km per millisecond. So the smallest possible time to ping Arizona State University with my packet travelling at a speed of light is  20000/300=67ms   I have noticed similar results for servers located in Australia and some other States of America. Is my internet connection five times faster than light? Analyzing limits problem Calculus (tell me where I'm wrong). I came accross: $$\lim_{x \to\ 0}\frac{x\cos x - \log (1 + x)}{x^{2}}$$  I tried the following please tell me where I 'm wrong:  $$\lim_{x \to\ 0}\frac{x\cos x - \log (1 + x)}{x^{2}} $$  $$\text{(Dividing by }x)$$   $$=\displaystyle\lim_{x \to\ 0}\dfrac{ \cos x - \dfrac{\log (1 + x)}{x}}{x} $$  $$=\lim_{x \to\ 0}\frac{\cos x - 1}{x} $$  $$=\lim_{x \to\ 0}\frac{-2\sin^{2} \dfrac{x}{2}}{x} $$  $$=\lim_{x \to\ 0}\frac{-2x\sin^{2}\dfrac{x}{2}}{(\dfrac{x}{2})^{2}\times 4} $$ $$\lim_{x \to\ 0}\dfrac{-x}{2}=0$$ But Answer given is $\dfrac{1}{2}$  Please Help.