If $b_{n+1}=\sqrt{2+b_{n}}$ for $n=1,2,3,\cdots\cdots$. Then $\lim_{n\rightarrow\infty}b_{n}$ is , given $b_{1}=3$   Try: let $b_{n}=2\cos(x_{n})$. Then put into $b_{n+1}=\sqrt{2+b_{n}}$  We have $$2\cos(x_{n+1})=2\cos\bigg(\frac{x_{n}}{2}\bigg)$$  Could some help me how to calculat $\lim_{n\rightarrow\infty}b_{n}$  Thanks in advanced
(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: .
How do I upgrade to a newer version of Ubuntu? prove that : $\cos x \cdot \cos(x-60^{\circ}) \cdot \cos(x+60^{\circ})= \frac14 \cos3x$ I should prove this trigonometric identity. I think I should get to this point : $\cos(3x) = 4\cos^3 x - 3\cos x $ But I don't have any idea how to do it (I tried solving $\cos(x+60^{\circ})\cos(x-60^{\circ})$ but I got nothing) How can I get a DisplayLink USB monitor work under Ubuntu? I have a DisplayLink-based external USB monitor, which has both power and data over USB, and seems to work perfectly under Windows 7, but only can display a text console under Ubuntu 10.10, and that I can only use when I am actually switched to it. So the only Use I can have so far is to have some text-based monitoring or console that I can watch while working on the graphic display.   I know there are some development done for DisplayLink, but I never could get it to actually run properly.   Yes, I tried the detailed setup described in other , but they did not work, and instead crashed my X that I had to restore.   Merci :-) Set value of textarea in jQuery I am attempting to set a value in a textarea field using jquery with the following code:  $("textarea#ExampleMessage").attr("value", result.exampleMessage);   The issue is, once this code executes, it is not altering the text in the textarea?  However when performing an alert($("textarea#ExampleMessage").attr("value")) the newly set value is returned? I am looking to sell and buy a house. But with the current market being as crazy as it is (houses are selling fast for sometimes 150,000 over asking), I'm afraid if we sell first, we could be stuck without a new house. Or stuck settling on or over-paying for something and wishing we never sold at all. I'm wondering if it's possible to buy first. What is the best way to secure my next house before selling my current house without getting locked into a B-lender mortgage or risking too much?  I know there would be some risk putting an offer on a new house without knowing how much the current house will sell for, or the risk we don't sell the current house before the new house closes. I'm also afraid conditional offers will not be considered. Are there any other options?  (We do have a mortgage on the current house, but because the market has gone up so much there is also a good chunk of equity.) This seems to be a basic issue, but I just realized that I actually don't know how to test equality of coefficients from two different regressions. Can anyone shed some light on this?   More formally, suppose I ran the following two regressions:  $$ y_1 = X_1\beta_1 + \epsilon_1 $$ and  $$ y_2 = X_2\beta_2 + \epsilon_2 $$ where $X_i$ refers to the design matrix of regression $i$, and $\beta_i$ to the vector of coefficients in regression $i$. Note that $X_1$ and $X_2$ are potentially very different, with different dimensions etc. I am interested in for instance whether or not $\hat\beta_{11} \neq \hat\beta_{21}$.   If these came from the same regression, this would be trivial. But since they come from different ones, I am not quite sure how to do it. Does anyone have an idea or can give me some pointers?   My problem in detail: My first intuition was to look at the confidence intervals, and if they overlap, then I would say they are essentially the same. This procedure does not come with the correct size of the test, though (i.e. each individual confidence interval has $\alpha=0.05$, say, but looking at them jointly will not have the same probability). My "second" intuition was to conduct a normal t-test. That is, take   $$ \frac{\beta_{11}-\beta_{21}}{sd(\beta_{11})} $$  where $\beta_{21}$ is taken as the value of my null hypothesis. This does not take into account the estimation uncertainty of $\beta_{21}$, though, and the answer may depend on the order of the regressions (which one I call 1 and 2).   My third idea was to do it as in a standard test for equality of two coefficients from the same regression, that is take  $$ \frac{\beta_{11}-\beta_{21}}{sd(\beta_{11}-\beta_{21})} $$   The complication arises due to the fact that both come from different regressions. Note that   $$ Var(\beta_{11}-\beta_{21}) = Var(\beta_{11}) + Var(\beta_{21}) -2 Cov(\beta_{11},\beta_{21}) $$ but since they are from different regressions, how would I get $Cov(\beta_{11},\beta_{21})$?  This led me to ask this question here. This must be a standard procedure / standard test, but I cound not find anything that was sufficiently similar to this problem. So, if anyone can point me to the correct procedure, I would be very grateful! What does this red icon on my panel mean? This red icon.... It showed up after I upgraded to Ubuntu 12.04LTS; I can't figure out how to ditch it. It seems to be a notification thing.  I am at a loss.  The icon is this one:      I clicked it and read this:     An error occurred, please run Package Manager from the right-click menu or apt-get in a terminal to see what is wrong. The error message was:  'Error: BrokenCount&gt;0'.        This usually means that your installed packages have unmet dependencies. How do I simulate in Photoshop what a piece will look like when printed on non-white paper? I'm designing a box in Photoshop that will be offset printed on unbleached paper. I haven't picked out a printer yet, so I don't know the exact kind of paper I'll be using and don't have access to it for proofs. For now I'm using a lightly-textured layer of tan color as a placeholder to mock it up.  What I'm trying to figure out is, how do I figure out how my design will look printed on natural paper rather than bleached white paper? I know that the brightness will be diminished and the hue will be altered, but I need to know how much so I can compensate for it as much as possible.  Is there a way to get at least a rough but reliably-simulated idea of how my prints will come out? Does it have to do with blend modes, or color separations? I'm at a bit of a loss here. Showing $F[x,y]/(ax^2+by^2-1)$ is a Dedekind domain What is the purpose of the three part circuit? I'm studying for my final on introduction to circuits and came across a question about this circuit:  The question is:  Identify and describe the three parts of the circuit used for a measurement system for signal conditioning. Describe the signal conditioning one intends to use the circuit for.  From what I can tell the R1 and R2 resistors are the first part of the circuit used to use a fraction of the incoming voltage. The second part is the voltage follower used to move all of the incoming voltage to another circuit. Lastly there is a low pass filter used to filter off the high frequency voltage. The second question I'm not sure of the answer. In the Lord of the Rings it is said     "In Eregion long ago many Elven-rings were made, magic rings as you   call them, and they were, course, of various kinds: some more potent   and some less. The lesser rings were only essays in the craft before   it was full-grown, and to the Elven-smiths they were but trifles — yet   still to my mind dangerous for mortals. But the Great Rings, the Rings   of Power, they were perilous.”   [LotR I 2 (60)]   There is a clear distinction here between the 20 Rings of Power in the Ring Rhyme (the 7,9,3,and 1) and these other rings. What happened to the "lesser rings"? Can a union be initialized in the declaration? Access: Making a Pivot Table with Multiple, Same Values as Columns The Weyl equidistribution theorem states that the sequence of fractional parts ${n \xi}$, $n = 0, 1, 2, \dots$ is uniformly distributed for $\xi$ irrational.   This can be proved using a bit of ergodic theory, specifically the fact that an irrational rotation is uniquely ergodic with respect to Lebesgue measure. It can also be proved by simply playing with trigonometric polynomials (i.e., polynomials in $e^{2\pi i k x}$ for $k$ an integer) and using the fact they are dense in the space of all continuous functions with period 1.  In particular, one shows that if $f(x)$ is a continuous function with period 1, then for any $t$,  $\int_0^1 f(x) dx = \lim \frac{1}{N} \sum_{i=0}^{N-1} f(t+i \xi)$. One shows this by checking this (directly) for trigonometric polynomials via the geometric series.  This is a very elementary and nice proof.  The general form of Weyl's theorem states that if $p$ is a monic integer-valued polynomial, then the sequence ${p(n \xi)}$ for $\xi$ irrational is uniformly distributed modulo 1.  I believe this can be proved using extensions of these ergodic theory techniques -- it's an exercise in Katok and Hasselblatt.  I'd like to see an elementary proof.  Can the general form of Weyl's theorem be proved using the same elementary techniques as in the basic version? Every nonzero element in a finite ring is either a unit or a zero divisor Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor. How do I mimic Pathauto transliterated output for an alias in my custom PHP code? When writing my custom function I need to manually append node name to a taxonomy term link. To get the taxonomy term part done I use:    $link_uri = taxonomy_term_uri($term);   $path_alias = drupal_get_path_alias($link_uri['path']);   This works ok, now - I want to append node name ("/" . $node_name), but instead of raw string, I want the Pathauto transliterated output, as set in the URL alias configuration, for that node context.  What function should I use to mimic that output? I looked at hook_pathauto_alias_alter(&amp;$alias, array $context) but I'm not sure that's what I need. What does Redis do when it runs out of memory? is $\sqrt{x}$ always positive? Is it possible to play an animation in the viewport at a slow framerate?  Just because of the rather heavy scene it's already going at about 8 fps, but I would like to be able to play it even slower (an in a controlled manner) for the purpose of recording an animation with automatic keyframes. UAE visit visa with expired passport number