Consider the recursive relation $ a_0=1, \ a_1=1 \ \ and \ \ a_{n+1}=a_n+a_{n-1} $.   Assuming that $ \ \ a=\lim |\frac{a_n}{a_{n+1}}| \ $ exists , find the limit $ \ a \ $.  Answer:  $ \ \ a=\lim |\frac{a_n}{a_{n+1}}|=\lim |\frac{a_n}{a_{n}+a_{n-1}}| $ , (since $a_{n+1}=a_n+a_{n-1}) ,   $  or, $ a=\lim |\frac{\frac{a_n}{a_{n-1}}}{\frac{a_n} {a_{n-1}}+1}| $   or, $ a=\frac{\lim|\frac{a_n}{a_{n-1}}|}{\lim|\frac{a_n}{a_{n-1}}+1|}=\frac{a}{\lim|\frac{a_n}{a_{n-1}}+1|}$ ,  (Since $ \lim |a_n /a_{n-1}|=\lim |a_{n+1}/a_n| $ ) .   Or, $ \lim|\frac{a_n}{a_{n-1}}+1|=1 $   But I can't proceed furher . Any help is really ppreciating .
I have read in a book that the limit of the ratio of consequent Fibonacci numbers is the golden ratio. However, it was just mentioned thus not justified. So, my question is how would you derive the following limit: $$\lim_{x\to\infty}{\frac{F_n}{F_{n+1}}}=?$$ Where $F_n$ is the nth Fibonacci number?
Why *not* parse `ls` (and what to do instead)? What do you call someone who can't keep secrets? Some one who is not good at keeping secrets. In my native language it is called "chugalkhor" but it's a slang. So I can't translate it.  What do you call such a person who can't keep secrets because his instincts don't let him. Are there any synonyms for the noun 'crush'? I must clarify, when I say 'crush', I mean the subject of an unrealistic, one-sided infatuation. For example, it can be used in this sentence: 'My celebrity crush is David Beckham.'  Could that word be interchanged for another? Preferably, can it be a word which is not necessarily too formal/complex? I've searched online for a while but can't find any synonyms for this form of 'crush.'  I want the word for a novel which is set in an old-ish fantasy period so I don't think 'crush' would feel in-place.  The main character is a lower-class worker. What was with the two different colored prison uniforms in Daredevil season 2? In Daredevil S02, we see two kinds of prison uniforms, full orange and full white. We see in the flashback that Wilson Fisk gets a white prison uniform but his current one was orange.  The Punisher was also given a white prison uniform but most of the prisoners had orange uniforms. What was with the two different colored uniforms? Does it have some significance or is it just random? Does it happen similar in real life US prisons too? Normality test with large data set I think I am dealing with this issue   I have large data set(10k data points) that slightly diverges from normal, and I get p-value of 0. I am interested in having perhaps a more crude test that tells me if data is extremely divergent from normal, versus looking somewhat normally distributed. I am currently trying Kolmogorov-Smirnoff, and in both cases I just get p-value of 0. Any alternatives?  I also looked at this:   So, taking all this into consideration, is there any kind of test I can perform that distinguishes between data that's roughly normal, and not normally distributed at all?  I am using scipy. Firstly, how do I read it? Is below right? With braces   $$p \vdash (q \Rightarrow p)$$  The given proof is:    ${p}$, premise ${q}$, assumption  $p$ by (1) // what???  ${q} \Rightarrow p$ by implication introduction with 2 and 2.1 QED...    I can't really get the link between $p$ and $q$ in 2.1 ... Want to set a guest session password for login. How do you do this after skipping it at OS install? QGIS locked into fullscreen mode on Ubuntu 12.04 How to read over 4k input without new lines on a terminal? So I have a lot of data WITHOUT NEW LINES on the clipboard (it's a large SVG file on one line). I went  $ cat &gt;file.svg   then tried to paste (in Gnome Terminal), but only the first 4kB characters were accepted.  I assume this is a readline feature/limitation.  Is there a way to read from STDIN that would avoid this problem?  EDIT  Test case: Create a demo file. This one will have ~4k "=" symbols followed by "foo bar".  { printf '=%.0s' {1..4095} ; echo "foo bar" ; } &gt; test.in   Copy that into your clipboard  xclip test.in   (if you want to middle-click to insert) or  xclip -selection clipboard test.in   (if you want to use Ctrl-Shift-Insert to past it in)  Then cat &gt;test.out, paste (whichever way). Press Ctrl-D to end the stream. cat test.out - do you see "foo bar"?  On my set-up (Ubuntu 12.04, Gnome Terminal, zsh) when I paste I only see the = and I don't see foo bar. Same when I inspect test.out. I currently have the Geometry Dash lite version and I want to buy the full version. Can I login with my account of Geometry Dash lite and keep everything I already had? It would suck if I had to redo everything. Issues Doing a String Comparison in LINQ Interpretation of the rating of a real transformer The rating of a real transformer is 15 kVA , 2300/230-V then can we say that the primary rms is 2300 V or can we say that the secondary rms is 230 V or both.  To clarify further, suppose to solve for certain characteristics of this transformer like primary current and secondary current i use the equivalent circuit equation of the transformer referred to the primary side as:  Vp = aVs + Req*Ip + jXeq*Ip  Then what should i use for Vp or Vs in the above equation looking at the rating of the transformer.  In the above equation looking at the rating data given above is Vp= 2300 or Vs = 230 ?As mentioned earlier the transformer is mentioned to be real. Linux / Bash, using ps -o to get process by specific name? How do I parse an ISO 8601-formatted date? I need to parse  strings like "2008-09-03T20:56:35.450686Z" into Python's datetime type.  I have found  in the Python standard library, but it is not very convenient.  What is the best way to do this? While I applaud the excellent work of , the question has come up:     How long will be the elected moderator’s “mandate”? Is it for life, or are we going to have “moderator elections” regularly?   Of course, any community moderator who wishes to recuse him or herself can certainly do that at any time.   Should we have term limits that require community moderators be cycled out periodically, so others get a chance to moderate? Should we hold regular (say, yearly) moderator elections? Is it better to have a larger pool of community moderators, and let the normal ebb and flow of turnover get us "natural" variability in moderation?   What are your thoughts on this? How much energy does a photon need to form a black hole? I was wondering if it is possible for a single photon to form a black hole if it has a small enough wavelength.If so, what would this wavelength be? I came across this question because I am reading about loop quantum gravity, and have heard that it is 'untestable'. I know that the loops making up spacetime in loop quantum gravity are thought to have a size in the order of $10^{-35}$m, and I was wondering whether making a photon with a wavelength small enough to investigate this would form a black hole.  I am aware of the two posts here, but neither seem to really answer my question because they refer to several photons:  and .  Thank you. Date comparison : which date format? I searched here and see mention of Clonezilla and Remastersys - but nothing I found answered my question completely, so I ask:  My home PC is just a generic one, and now that I changed it to Ubuntu I really love how it's come out and how it runs better !  My office PC is a different generic set of hardware, and I'd like to 'clone' the exact same setup to it but not have to repeat all the tweaking and such that I did already.  Is there a program that will allow me to replicate/install my already customized setup from one to the other, easily ? Are there certain steps I must make before doing something like that so it'll be compatible with different hardware ?  Thanks ! Display maximum possible count of sites in footer Could a constant function be considered increasing? Given the definition of an increasing function as a function $f:E\to\mathbb{R}$, where $E$ is nonempty such that $x_1,x_2\in E$ and $x_1&lt;x_2$ imply $f(x_1)\leq f(x_2)$, then couldn't, for example, $f(x)=-1$ considered increasing? Since the inequality is not a strict inequality.