Let $\{ a_{n}\}$ be a positive real sequence $a_{n}=\sqrt{a_{n-1}a_{n-2}}$ for $n≥3$, then $\{ a_{n}\}$ converges to $(a_{1}a_{2}^{2})^{1/3}$.
Limit of a sequence defined by recursive relation : $ a_n = \sqrt{a_{n-1}a_{n-2}}$
Bottom frame of listing appears on top of the next page Turning off "Update page numbers only" in Word When I update the fields document-wide (Ctrl + A, then F9) in a Word document that has a table of contents, a dialog appears asking if I want to update page numbers only, or update the entire table. I never want to update page numbers only if the document headings have changed, nor will I ever.  (Apparently this is given as an option in case someone has manually changed an automatically generated ToC, which is almost always a really bad idea.)   Unfortunately, "Update page numbers only" is the default selection in the dialog, which means that I have to select the other option every time. I update the fields all the time when I'm working in a document that has figures and cross-references, so this is an annoying extra step I'd really like to get rid of.   Is there any way to tell Word to stop asking me this stupid unnecessary question? If $x>0$, $\,x^{1/n}$ tends to $1$ as $n\to\infty$ I am looking for a proof possibly using the sandwich theorem and/or Bernoulli's inequality for proving the following statement:  If $x&gt;0$ then $x^{1/n}$ tends to $1$. How to permanently set environmental variables What do I need for extreme macro photography? Short sci-fi story of plan devised by marooned spaceship pilot to save paradise world from development Spaceship crashes on beautiful, remote planet. The pilot survives and lives quite well on the beautiful "island paradise" world amongst friendly, good-natured people... but, one day (while relaxing in a hammock near a beach) he "realized he was dying" (actually, that he would eventually die, then being unable to help the people he had come to love). He fears future "discovery" of his adopted world and the inevitable plundering of the resources and people by "developers"... so he devises... "a plan" "convinced island elders to support his plan" "his students taught others" "achieved 80% literacy... a remarkable feat" "established world government" "more generous than developer expected" "successfully argued for independent status" "established 100% taxation rate" "finished resort buildings converted to school" "developer hired to build resort in isolated area"  ...'not having luck finding it... my recollection of the story is good, but apparently not EXACT/searchable phrases How do I get the value of text input field using JavaScript? Possessive of a word that is already possessive If the cricket ground Lord's is a possessive, what if you want to describe something belonging to Lord's? Would you say:     I was very impressed by Lord's's customer services.   It doesn't look right, so what is the correct way of writing it? Fill the area determined by two pgfplots graphs I need to fill the region determined by two (Gaussian) curves and the right red line (the region is brushed in green in the figure below). I also wonder what is the best way to draw vertical line which goes through the intersection of both graphs (the left red line on the figure).  Any pointers would be greatly appreciated.    My initial attemt is pasted below.  \documentclass{standalone} \usepackage{pgfplots}  \begin{document}  \pgfmathdeclarefunction{dnorm}{2}{%   \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}% }  \begin{tikzpicture} \begin{axis}[domain=0:12, samples=100, height=5cm, width=10cm] % Fill aread under the curves \addplot [fill=red!20, draw=none, domain=0:6] {dnorm(6.5,1.5)} \closedcycle; \addplot [fill=blue!20, draw=none, domain=6:10] {dnorm(4,1)} \closedcycle; % Draw curves \addplot [thick] {dnorm(4,1)}; \addplot [thick] {dnorm(6.5,1.5)}; \end{axis} \end{tikzpicture}  \end{document}   My solution  A picture is worth a thousand words. Code is pasted below.   \documentclass{article} \usepackage{tkz-fct} \usetikzlibrary{intersections}  \begin{document}  \tikzset{    name plot/.style={every path/.style={name path global=#1}} }  % Extract coordinates for point X \makeatletter \newcommand{\gettikzxy}[3]{%   \tikz@scan@one@point\pgfutil@firstofone#1\relax   \edef#2{\the\pgf@x}%   \edef#3{\the\pgf@y}% } \makeatother  % Dimlines \def\Dimline[#1][#2][#3][#4]{ \begin{scope}[thin, &gt;=stealth'] % redefine as flechas \draw let \p1=#1, \p2=#2, \n0={veclen(\x2-\x1,\y2-\y1)} in [|&lt;-&gt;|, decoration={markings,mark=at position .5 with {\node[#3] at (0,0) {#4};}, }, postaction=decorate] #1 -- #2 ; \end{scope} }  \begin{tikzpicture}[scale=1,font=\small] \tkzInit[xmin=0,xmax=12,ymin=0,ymax=.3,ystep=.05] % Draw coordinates \draw[&gt;=stealth', &lt;-&gt;] (0,6) node[above] {$y$} -- (0,0) --  (12.5,0) node[right] {$x$}; % Draw functions and areas \tkzFct[name plot=A,thick,color=red,domain=0:12]{1/(1.5*sqrt(2*pi))*exp(-((x-4.5)**2)/(2*1.5**1))} \tkzDrawArea[opacity=.3,color=blue,domain = 7:12] \tkzFct[name plot=B,thick,color=blue,domain=0:12]{1/(2*sqrt(2*pi))*exp(-((x-7)**2)/(2*2**1))} \tkzDrawArea[opacity=.3,color=red,domain=0:7] \tkzDrawAreafg[between=b and a,opacity=.3,color=green,domain = 0:7] % Intersection between curves \path [name intersections={of=A and B,by=C}]; % Extract coordinates of C \gettikzxy{(C)}{\cx}{\cy} % Vertical lines \draw [thick,dashed, black] (\cx,0) -- (\cx,5.5) node [above] {$x_{0}$}; \draw [thick,dashed, black] (7,0) -- (7,5.5) node [above] {$\hat{x}$}; % Define regions \Dimline[($(0,0)+(0,-.6)$)][($(7,0)+(0,-.6)$)][above,black][$\mathcal{R}_{1}$]; \Dimline[($(7,0)+(0,-.6)$)][($(12,0)+(0,-.6)$)][above, black][$\mathcal{R}_{2}$]; \end{tikzpicture}  \end{document} Can I use mirroring to make a copy of my Database? printing a block of text that matches a pattern Using just shell script, how to search a text file and list all whole blocks of lines that has inside some text (simple grep criteria).   The text file has blocks of lines separated by "-----------------" (precisely, each block start with "\n\n\n--------------------"... about 50 chars "-").  A sample could be:  ------------------------------- Abracadabra, blablablalbalba blablablabla, banana    ------------------------------- Text, sample text, sample text, sample text Text, sample text, sample text, sample text Text, sample text, sample text, sample text Text, sample text, sample text, sample text   -------------------------------   Text, sample text, sample text, sample text banana. Sample text, sample text, sample text, sample text Text, sample text, sample text, sample text   Lets consider the word "banana" the search criteria. So, the blocks listed would be:  ------------------------------- Abracadabra, blablablalbalba blablablabla, banana   ------------------------------- Text, sample text, sample text, sample text banana. Sample text, sample text, sample text, sample text Text, sample text, sample text, sample text   EDIT:  Testing answers to try awk, like: awk 'BEGIN{RS="\n------------"}/INFO/{print}' where INFO is what was searched for. I cannot get the whole block. So, follows a real sample and the result:  A REAL SAMPLE (including the first 3 new lines):      ------------------------------------------------- Diretório separado do nome o arquivo: adis, IWZLM (/home/interx/adis/src/IWZLM.SRC) Gerando rotina em linguagem C: (yla5 adis IWZLM -if) .INFO =>Rotina BLOQUEADA (status 'M'): Geracao ignorada (use -is para ignorar checagem do status)  [  OK-I ] IWZLM (adis) - Lista lay: Geracao ignorada do codigo em C.    ------------------------------------------------- Diretório separado do nome d arquivo: adis, ADISA (/home/interx/adis/src/ADISA.SRC) Gerando rotina em linguagem C: (yla5 adis ADISA -if) .ERRO: Falha inesperada  Compilando o programa: (ycomp adis ADISA -exe adis/exe/ADISA.temp.exe ) adis/exe/ADISA.temp.exe => adis/exe/ADISA  [  OK   ] ADISA (adis) - Menu A : Gerada e compilada com sucesso.    ------------------------------------------------- Diretório separado do nome o arquivo: adis, ADISD1 (/home/interx/adis/src/ADISD1.SRC) Gerando rotina em linguagem C: (yla5 adis ADISD1 -if) .ATENCAO: Definicao nao localizada  Compilando o programa: (ycomp adis ADISD1 -exe adis/exe/ADISD1.temp.exe ) adis/exe/ADISD1.temp.exe => adis/exe/ADISD1  [  OK   ] ADISD1 (adis) - Menu : Gerada e compilada com sucesso.   I cannot get the whole block, just the line containing "INFO", like a ordinary grep, either setting or not ORS:  $ cat file  | awk 'BEGIN{RS="\n------------"}/INFO/{print}'  .INFO =&gt;Rotina BLOQUEADA (status 'M'): Geracao ignorada (use -is para ignorar checagem do status)   NOTES: It is the awk from AIX 7.1, not gawk. How to change the catcode in a macro Latin Square and Graph Theory How many &quot;$1$&quot; can be selected at most, provided that only one from each row and column of the attached matrix $A$ is selected?(That is, when one &quot;$1$&quot; is selected, no other element can be selected than the row and column where that &quot;$1$&quot; is located.)  Answer: I think $6$ can be chosen.  How can I give this answer in connection with graph theory? I think it has to do with the Latin Square(or Rectangle). But I can't connect and explain. Please help me. I thought a lot, but I could not establish the relationship completely. Thanks... I sold an investment in a standard brokerage account at a small loss because I needed the money for a down payment.  I still think it is a good investment, can I buy the same investment in a standard IRA without running into wash sale problems?  I will probably just wait the 30 days any way to avoid any headaches, but I want to know if there is a definitive answer on this.    Being as the cost basis in the traditional IRA doesn't matter, I wouldn't think there would be a problem. Monitor hard drive changes to files Several months ago I saw someone using an application that showed him files on the computer being modified in real time.  Basically there was an application that wasn't working correctly, so he installed the program, ran the application, and saw all the various files that the application either read from or wrote to.  He found out the application was writing a large debugging file it shouldn't be, which ended up being the problem.  I don't remember the application or the name of the person who used it, but I believe it's a freeware application.  Does anyone know what this program might be? I have a function defined as follows: $f(x,y)= \dfrac{x^2-y^2}{\left(x^2+y^2\right)^2}$, if $(x,y)\neq (0, 0)$ and $f(x,y)=0$ if $(x,y)=(0,0)$.  Now, $$\int_0^1\int_0^1 f(x,y)\,\text{d}x~\text{d}y=-\frac{\pi}{4}$$ and $$\int_0^1\int_0^1 f(x,y)\,\text{d}y~\text{d}x=\frac{\pi}{4}.$$   The question I have is this: why does this not contradict Fubini's theorem?    thanks. I'm self-studying Guillemin and Pollack, but I'm stuck on Problem 3 of section 2. It says that if $V$ is a vector subspace of $\mathbb{R}^N$, then $T_x(V)=V$ if $x\in V$.  If $x\in V$, then since $V$ is a manifold, there is a local parametrization $\phi: U\to V$ where $U$ is open in $\mathbb{R}^k$. Without loss of generality, we can require $\phi(0)=x$. Then $T_x(V)$ is defined to be the image of $d\phi_0$ on $\mathbb{R}^k$.   An arbitrary element of $T_x(V)$ looks like $$ d\phi_0(v)=\lim_{t\to 0}\frac{\phi(0+tv)-\phi(0)}{t}=\lim_{t\to 0}\frac{\phi(tv)-x}{t} $$  but this doesn't seem very useful to show $T_x(V)=V$. What is the right approach? Where should I copy-paste-install software to? I'm installing IntelliJ, and the readme tells me to choose an installation folder. Where is a good, standard place to put installed programs? Links that are italics and bold not showing as links in Mobile Web Brightness is reset to maximum on every restart The brightness of my laptop is reset to max on every restart. I tried the solution provided at  but had no luck.   This command   cat /sys/class/backlight/acpi_video0/max_brightness   Returns   cat: /sys/class/backlight/acpi_video0/max_brightness: No such file or directory   Then I found that I don't have a folder named acpi_video0, but a folder called intel_backlight:    Every time I increase or decrease the brightness using the brightness control keys, the values in brightness and actual_brightness get updated.  Is there any method I could follow to set the brightness to a fixed value on every boot and vary it as and when I need it using the brightness control keys?