Changing order of integration in Double Integral
Evaluate the integral $\int_0^{2a}\int_0^\sqrt{2ax-x^2}\frac{\phi'(y)(x^2+y^2)x dxdy}{\sqrt{4a^2x^2-(x^2+y^2)^2}}$
If a series converges, then the sequence of terms converges to $0$. How do I change folder timestamps recursively to the newest file? Why does a star beyond a certain mass limit (Chandrasekhar limit) only become a black hole? Why does a star beyond a certain mass limit () only become a black hole?   A star is first made of hydrogen, it undergoes nuclear fussion reaction combining into helium and releasing a large amount of energy. This process continues till star is made up of iron core as iron has largest value of binding energy per nucleon, after this if mass of the star is above the value of Chandrasekhar limit it becomes a black star, what is the reason for this and why is certain mass limit required? Does the series $\sum \sin^{(n)}(1)$ converge, where $\sin^{(n)}$ denotes the $n$-fold composition of $\sin$? Can I downgrade to an earlier version of Android after installing an OTA update? I installed the KitKat upgrade on my Nexus 7, but I am unhappy with it. Is it possible to roll back to an earlier version of Android (Jelly Bean)? Uniform convergence of $f_n(x)=nx^n(1-x)$ for $x \in [0,1]$? I want to decide whether or not $f_n(x)=nx^n(1-x)$ is uniformly convergent or not. I have shown that $\lim_{n\to\infty} f_n(x)=0$ for $x \in [0,1]$.  Now $f_n(0)=f_n(1)=0$. And in $(0,1)$, we have $f'_n(x)=n^2x^{n-1}-n(n+1)x^n$. Putting this to zero gives $x=\frac{n}{n+1}$. Also $f_n(\frac{n}{n+1})=(\frac{n}{n+1})^{n+1}$. But that tends to $\frac{1}{e}$ as $n$ tends to $\infty$ which is not $0$ so we don't have uniform convergence, right?  Nonetheless we have $\lim_{n\to\infty} \int_0^1 f_n(x)dx=\lim_{n\to\infty} (\frac{n}{n+2}-\frac{n}{n+1})=0=\int_0^1 \lim_{n\to\infty} f_n(x)dx$.  Is all this right? I have some doubts, I mostly doubt my intuition in analysis... Thank you! Movie Clip On Plane In blender cycles render engine, how do I put a movie clip on a plane so it displays the video? I want the video to be on the plane if there is a way. I have two questions. First, which command lists files and directories, but lists directories first?  Second question: I want to copy a list of files into a single directory, but make the target directory the first filename in the command. Can anyone explain precisely what IOWait is? Objects not rotating or scaling in viewport Like the title says, can a matrix in $\mathbb{R}$ have a minimal polynomial that has coefficients in $\mathbb{C}$?  I have a feeling the answer is yes because the characteristic polynomial can have coefficients in $\mathbb{C}$ and so I don't see any reason why not, I was merely asking in case there was some sneaky proof that proves that this cannot happen. I have added the png transparent image with the setting and its not showing transparent black background is seen. I have the base color aluminum and i want to add that logo of png on to it. The image is black letter with transparent background why its showing in white letters and black background. May i know where im going wrong any suggestion or help. Do we have different set of badges in different SE sites. eg. I don't see any 'Pundit' badge in badges popup in Hinduism SE.  Also, if the above is true, then can we include the left badges in a SE site at a later stage. I'm trying to do something similar to this epsilon-delta graph I've found.  Specifically, I'd like to copy the placements and arrows, like the L+e and L-e coming out of L. Same thing with the x0 + d and x0-d coming out of the x0. Can someone help me with this? This is what I have so far.    \begin{tikzpicture} \begin{axis}[ ,         legend pos=north east,         axis x line=middle,         axis y line=middle,         x tick label style={/pgf/number format/fixed,                             /pgf/number format/fixed zerofill,                             /pgf/number format/precision=1},         y tick label style={/pgf/number format/fixed,                             /pgf/number format/fixed zerofill,                             /pgf/number format/precision=1},         grid = major,         width=8cm,           height=6cm,         grid style={dashed, gray!30},         xmin=0   ,     % start the diagram at this x-coordinate         xmax= 4,    % end   the diagram at this x-coordinate         ymin= 0,     % start the diagram at this y-coordinate         ymax= 11,   % end   the diagram at this y-coordinate %axis background/.style={fill=white},         xlabel=$x$,         ylabel=$f(x)$,         tick align=outside,         enlargelimits=false] % plot the stirling-formulae %\addplot[domain=0:200, red, ultra thick,samples=100] {K/(1 +    ((K-X0)/X0) * exp(r*x))    }; \addplot[domain=0:200, red, ultra thick,samples=100] {3*x-1 };  \addplot[line width=1pt, dashed, mark=none, black] coordinates {(0,8) (3,8)}; \addplot[line width=1pt, dashed, mark=none, black] coordinates {(3,0) (3,8)}; \addplot[line width=1pt, dashed, mark=none, blue] coordinates {(3.33,0) (3.333,9)}; \addplot[line width=1pt, dashed, mark=none, blue] coordinates {(2.666,7) (2.666,0)}; \addplot[line width=1pt, dashed, mark=none, blue] coordinates {(2.666,7) (0,7)}; \addplot[line width=1pt, solid, mark=none, blue] coordinates {(3.333,9) (0,9)}; \addplot[holdot] coordinates{(60,8.66)};  % 1\addlegendentry{$f(x) = 10 \sin (x)$}  \end{axis}   \end{tikzpicture}   which gives Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$ Prove that for any natural number $n&gt;1$, we have $(2n)!&lt;(n(n+1))^n$    I tried using induction, but I failed in that approach, I rather found it was untrue, but on several case testing, I found it to be true. I am out of ideas here, please help. Thank you. New Beamer Version: Transparent Title Page Boxes Consider the following file:  \documentclass{beamer}  \setbeamercolor{title}{bg=red} \setbeamercolor{author}{bg=yellow} \setbeamercolor{date}{bg=} \setbeamercolor{institute}{fg=red,bg=green}  \title{TitelTitelTitel} \author{meiermueller} \institute{Institute} \date{lalala}  \begin{document}  \usebackgroundtemplate{\color{blue}\rule{20cm}{15cm}}    \begin{frame}    \titlepage  \end{frame} \end{document}   On one of my computers (using MiKTeX 2.9 and beamer 2015/01/05 3.36), the result is the following:    Note how, in particular, the date box has a transparent background  On my other computer, I installed a new MiKTeX version (2.9.6800), including beamer 2018/09/02 v3.52, and get this:    Here, the date field has a white background, and I cannot seem to get it to be transparent.   Apparently, the difference has some relation to the beamer version, or maybe some other change in the new MiKTeX version.  But what ist the reason, and how do I get transparent backgrounds? Expected value of the number of flips until the first head Suppose we flip a coin until we see a head. What is the expected value of the number of flips we will take?  I am pretty new to expected value, so I tried to evaluate it by multiplying the probability of each scenario with the number of flips it took to get there (like taking the arithmetic mean). This didn't work though, because of the infinite possibilities. I'm really confused, and if someone were to provide an answer, I would really appreciate it if they could go into detail. There is a kind of situation that seems to occur somewhat commonly, that goes something like:   Figure out some way to handle a particular situation do it regularly, and make a habit of it, as it seems to work OK … … and only much later actually realize that it very much isn’t OK, that it’s been doing harm all the time, and that it’s too late to fix anything.   Does there exist any good words or phrases for this kind of situation? Repeating something with (1/n)th chance of success n times Is there anything that can be said about how many attempts it will take to correctly guess a random number out of 1000 numbers? If the number wouldn't change the probability would just increase every guess, giving in the end a probability of 1.  If the number changes every guess tough, you just have a 1/1000th chance everytime. Can it be calculated how many attempts it would probably take, on average?  (I'm sorry, i'm not very skilled in probability maths) for normal user I get :     /usr/lib/lightdm/lightdm:/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/usr/local/games:/home/monty/google_appengine   which is actually the content of /etc/environment  For root I get:     /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin   what's the reason behind this and which file contains this line?