If $a_1=1,a_{n+1}^2-2a_na_{n+1}-a_n=0$ then sum of the series $\sum {a_n\over 3^n}$ Let $\{a_n\}$ be a sequence of positive real numbers such that $a_1=1,a_{n+1}^2-2a_na_{n+1}-a_n=0\,\,\forall n\ge 1$. Then show that $$1&lt;\sum_{n=1}^\infty {a_n\over 3^n}&lt;2$$. I got this problem from a question paper. Though it didn't asked for exact sum of the series , if you also manage to find that then please post it.
$\{a_n\}$ be a sequence such that $ a_{n+1}^2-2a_na_{n+1}-a_n=0$, then $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in... Let $\{a_n\}$ be a sequence of positive real numbers such that   $a_1 =1,\ \  a_{n+1}^2-2a_na_{n+1}-a_n=0, \ \ \forall n\geq 1$.   Then the sum of the series $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in...  (A) $(1,2]$, (B) $(2,3]$, (C) $(3,4]$, (D)$(4,5]$.   Solution attempt:   Firstly, we figure out what $\frac{a_{n+1}}{a_n}$ is going to look like. We get, from the recursive formula, $\frac{a_{n+1}}{a_n}=1+\sqrt{1+\frac{1}{a_n^2}}$ (remembering the fact that $a_n&gt;0$, the other root is rejected).   We know that, if $\lim_{n \to \infty}\frac{a_{n+1}}{a_n}&gt;1$, then $\lim a_n \to \infty$.   Further, $(a_{n+1}-a_n)= \sqrt{a_n(a_n+1)}&gt;0$. (Again, the other root is rejected due to the same reason).   Hence, $(a_n)$ increases monotonically.   Therefore, the largest value of $\frac{a_{n+1}}{a_n}$ is approximately $1+\sqrt{1+\frac{1}{1}} \approx 2.15$  Now, the sum can be approximated as $\displaystyle\frac{\frac{1}{3}}{1-\frac{2.15}{3}} \approx 1.3$ (In actuality, $\mathbb{sum}&lt; 1.3$).  So, option $(A)$ is the correct answer.   Is the procedure correct?   I have been noticing a handful of this type of questions (based on approximations) lately, and the goal is to find out where the sum / the limit of the sequence might lie.   Is there any "definitive" approach that exploits the recursive formula and gives us the value, or does the approach varies from problem to problem?
So, somehow over the weekend, my 1GB USB flash drive has managed to become write-protected.  There is no switch on the stick, so I'm concluding that something has corrupted at some point.  Obviously I can't format it or remove.  The output from dmesg | tail is:  Buffer I/O error on device sdb1, logical block 328168 lost page write due to I/O error on sdb1 sd 73:0:0:0: [sdb] Unhandled sense code sd 73:0:0:0: [sdb] Result: hostbyte=DID_OK driverbyte=DRIVER_SENSE sd 73:0:0:0: [sdb] Sense Key : Data Protect [current] Info fld=0x0 sd 73:0:0:0: [sdb] Add. Sense: Write protected sd 73:0:0:0: [sdb] CDB: Write(10): 2a 00 00 05 08 00 00 00 01 00 Buffer I/O error on device sdb1, logical block 328200 lost page write due to I/O error on sdb1   I think the second and final lines are a giveaway to corruption, but I don't know how to interpret this output.  Could someone help to see if this output gives any indication?  I usually eject the drive before removing it, but I may have been lax once or twice.  The data isn't critical, it's just bugging me.  Any suggestions on how to fix it (if the drive even can be) would be greatly appreciated. I have been using this command successfully, which changes a variable in a config file and then executes a Python script within a loop: for((i=114;i&lt;=255;i+=1)); do echo $i &gt; numbers.txt; python DoMyScript.py; done  As each DoMyScript.py instance takes about 30 seconds to run before terminating, I'd like to relegate them to the background while the next one can be spawned. I have tried what I am familiar with, by adding in an ampersand as below: for((i=114;i&lt;=255;i+=1)); do echo $i &gt; numbers.txt; python DoMyScript.py &amp;; done  However, this results in the below error: -bash: syntax error near unexpected token `;' How do I transfer my Pokemon Go account to another device (Android to Android)? I was having trouble playing the game on my android so I downloaded it to my wife's phone (Galaxy S6). I read some posts on how to get it running on my android and it worked but, my level 7 account is on my wife's phone.  How do I get it back to my android? When I sign out and back in it doesn't ask for any sign in info. I've tried deleting the app's data and even uninstalling and re-installing, but it still doesn't ask for sign-in info. Suppose $A,B,C$ are sets. Prove that $A\mathbin\triangle B\subseteq C \iff A\cup C=B\cup C$. Suppose $A,B,C$ are sets.  Prove that $A\mathbin\triangle B\subseteq C \iff A\cup C=B\cup C$.  I proved that for $x\in A\setminus B$, $x\in A\cup C$ and for $x\in B\setminus A$, $x\in B\cup C$. How do I show that $A\cup C=B\cup C$, and that $A\mathbin\triangle B\subseteq C \iff A\cup C=B\cup C$? I have enabled platform encryption at my DE and selected Case Subject, Description and Comments to be encrypted.  I have a created a sample case with these values:    And when I query from Developer Console it returns as follows:       How do I know, subject and description are encrypted at Rest? Is there any indicator which can confirm that this data is encrypted?   By the way, in case of File/attachment there is isEncrypted attribute is available which denotes the same as follows. phpinfo() and php -v shows different version of PHP I wanted to know which php version I am using so I wrote the standard script:   phpinfo();   Which gives me  PHP version 5.6.10   The correct PHP version needed for my application. When I tried in terminal:  php -i   or  php -v   It shows me:  PHP version 5.3.2   Which I don’t need. Then I tried with:  which php   It shows me:  /usr/bin/php   Now I am confused which version of PHP I am using. The files are uploaded to the hosting website and I cannot make any changes to the config files like httpd.conf or php.ini. I'm sure there must be a simple reason for this but I can't seem to be able to import a shp file or a feature class from another file or sde geodatabase into a Feature Dataset without getting an error.  The feature classes come in fine if I don't don't try to import them into a Feature Dataset.  The feature dataset definitely has the same projection as the feature classes I'm trying to import so I don't think that's the problem.  The Error I get is ERROR 000210 and just says Failed to convert and cannot create output and Failed to execute (CopyFeatures).    Any ideas what I may be doing wrong here.  Seems like something so simple.  I'm trying to use the Feature Datasets to better organize my data which I think is there purpose.   Update:  So as you can see below, Jason helped me find the answer.   As suggested by Matt below, I added a new Idea to the ArcGIS Ideas site.  Here is the link to that, so please promote this up if you agree a better error message is needed here: Why are you not able to declare a class as static in Java? Is there a word to describe indecisiveness between two things? Possible Duplicate:          Is there a word to describe indecisiveness, specifically between two things? I tried online dictionaries, but did not find anything. Where in the registry does Windows store with which program to open certain file types? After installing Adobe PDF reader, it opens files with the .PDF file extension. Ones' default web browser opens .html files and notepad opens .txt files, etc.  Where exactly in the registry are these filetype - program associations stored in the registry?   In other words, how does Windows know with which software/program it must open a certain file(type)? How do I see all foreign keys to a table or column? In MySQL, how do I get a list of all foreign key constraints pointing to a particular table? a particular column? This is the same thing as , but for MySQL. What antivirus programs are available for Ubuntu?  We previously used Symantec Endpoint Protection but it does not work in Ubuntu. This is a story that was read on BBC radio about thirty years ago, and this is about as right as I can remember it: The human pilot of a spacecraft is doing some reconnaissance on an alien planet. He crashes but survives. For some reason, no one will know where to look for him. He sees something in the distance that looks like a building of some sort. When he gets there, it seems to be a home. Very much like a human home. There's furniture, but it's strange and uncomfortable . There's food, but it's inedible. There's a faucet but what comes out of it seems more like acid. Everything looks as though it should be right, but it's not. He leaves the house, wanders and after a few days is desperate for water. But there isn't any water anywhere. He's close to death, crawling on the ground and then something happens. There are cracks in the soil and there appears to be water bubbling up from perhaps some underground stream. The water revives him. He goes back to the house and tries the food again. It's delicious and he eats enough to gain back his strength. Then he decides to try the water again. It's pure and wonderful. He takes a shower and the last line of the story is how refreshing the water is on his skin and something about him lifting his tail. I'm attempting to cut a hole in a plane. I added a cylinder object, added the Boolean modifier to the plane, selected "cylinder" and "Difference", but no hole is made. The lower half of the cylinder just highlights, moving/deleting the cylinder reveals a hole-less plane.      How can i cut a hole in the plane? I apologize if this question is posted on the wrong portion of the website, but I knew no better place to ask.  I've come to see that some effects, such as super-conduction may happen at either extremely low temperatures, or at extreme temperatures (the recent discovery of superconducting iron in the bowels of the earth, for one), and this is fascinating.  My question is: can (does?) the same principle be applied to quantum effects? Would it be applicable to, for example, say that although quantum effects happen at nano scales and below they may happen on a grand scale of say, the entire Universe (as a solitary unit)?  Can said quantum effects happen at moments of say, picoseconds after the Big Bang due to the turbulence of forces involves in such an event? Or even within the core of a dying star?  Thank you for any answers or clarification of my misinformation. If $I_n=\int _0^{\pi }\sin^{2n}\theta \:d\theta $, show that $I_n=\frac{\left(2n-1\right)}{2n}I_{n-1}$, and hence $I_n=\frac{\left(2n\right)!}{\left(2^nn!\right)2}\pi $  Hence calculate $\int _0^{\pi }\:\sin^4t\cos^6t\:dt$  I knew how to prove that $I_n=\frac{\left(2n-1\right)}{2n}I_{n-1}$, but I am not very good at English, what does it mean Hence $I_n=\frac{\left(2n\right)!}{\left(2^nn!\right)2}\pi $ do we need to prove this part as well or is it just a hint to use? and for the other calculation to find $\int _0^{\pi }\:\sin^4t\cos^6t\:dt$ is there something in the first part that I can do to help me solve this question because otherwise it becomes very long and it's part of the question. Prove that every terms of a sequence defined by a recurrence relation is a perfect square. I am reading a paper and have questions about the details of the procedure described. I have read other papers by the same team but they don't explain too much about that procedure anyway. I think it might be common, but my supervisor doesn't know it too.  I am stuck and I want to get out of it. As a student working on my master thesis, can I email the contact author for the manuscript cold out, or should I ask someone to contact him for me? I would ask my supervisor but I don't want it to imply that I'm avoid taking initiatives when I could do it on my own. I've already read all the pages in this site trying to find the answer to my problem but no one seems to be the right one form me...  First I explain you the kind of data I'm working with...  Let's say that I have an array vector with several names of city, one for each of 300 users. I also have another array vector with scores response to a survey of each user or a continuous value for each user.  I would like to know if exist a correlation coefficient that compute the correlation between these two variables so, between a nominal and a numeric/continuous or ordinal variables.  I've searched on the Internet and in some pages they suggest to use the contingency coefficient or Cramer's V or Lambda coefficient or Eta . For each of this measure the just say that they could be applied for such data in which we have a nominal variable and interval or numerical variable. The thing is that searching and searching, trying to understand every one of them, sometime is written or watching the examples that they are reasonable to use them if you have dichotomous nominal variable, except for Cramer's V, other time is not written any requirement for the type of data. A lot of other pages say that is right to apply regression instead, that is right, but I would just simply like to know if there is a coefficient like pearson/spearman for this kind of data.  I also think that is no so properly to use Spearman Correlation coeff since the cities are not sortable.  I have also built the function of Cramer'sV and Eta by myself (I'm working with Matlab) but for Eta they don't talk about any p-value to see if the coefficient is statistically significant...  In the matlabWorks site there is also a nice toolbox that says to compute eta^2 but the kind of input it needs is not understandable.  Is here someone that have done a test like mine? If you need more detail to understand the kind of data I'm using just ask me and I'll try to explain you better. Execute script after specific delay using JavaScript Is there any JavaScript method similar to the jQuery delay() or wait() (to delay the execution of a script for a specific amount of time)?