Find the value of   $\displaystyle \lim_{n \to \infty} \int_{0}^{1} nx^n e^{x^2}dx$ I am trying to solve this by using uniform convergence rule for integration but getting nothing , i have solved questions with negative exponential ,because it is easy to find their uniform convergence ,but here i am not getting any idea how to solve this, please  give me a hint so that i can try it myself, Thank you.
The value of $\lim_{n \to \infty}\int_{0}^{1}nx^ne^{x^2}dx$ is _____ I tried by taking the odd $n$ values as in that case, the integral I suppose was easier to calculate. So, denote $I_n=n\int_{0}^{1}e^{x^2}x^ndx$, then we have : $I_1=\frac{e-1}{2},\\I_3=\frac{3}{2},\\I_5=5(\frac{e}{2}-1),\\I_7=7(3-e)$ Then I tried calculating (random) values using calculator as integration was getting cumbersome. $I_{31}=2.488$, $I_{51}=2.57$. I do not see any kind of recurrence so that I can find a general term for odd $n$. I also tried the method given . The idea was that since integral doesn't depend on $n$ considering $I(n)=\int_{0}^{1}e^{x^2}x^ndx$, then $I'(n)=\int_{0}^{1}x^n\ln(x)e^{x^2}dx$, but this also doesn't lead me to any conclusion as well. I believe I couldn't get to a proper approach to tackle this question. Can someone please help with the idea that involves in solving these type of questions?
Image viewer without large window borders I am looking for an image viewer software without very large window borders. I don't need any type of toolbar, just a simple image viewer.   It would also be fine if there is no top bar to close, minimize or maximize window. XeLaTeX update [20201004] and item ball in Beamer Beamer item balls go missing while compiling in XeLateX since yesterday's TeXlive update, however, the same code works with pdfLaTeX. Any solution, please. Here is my MWE: \documentclass{beamer}  \setbeamertemplate{itemize item}[ball] \setbeamertemplate{enumerate item}[ball]  \begin{document}  \begin{frame} \frametitle{A tested title} text \begin{enumerate}     \item this is an item \end{enumerate} text3 \begin{itemize}     \item this is an item \end{itemize} \end{frame} \end{document} How can I add custom '.desktop' application launchers to Ubuntu Dock? How to run long time process on Udev event? I want to run a ppp connection when my USB modem is connected, so I use this udev rule:  ACTION=="add", SUBSYSTEM=="tty", ATTRS{idVendor}=="16d8",\     RUN+="/usr/local/bin/newPPP.sh $env{DEVNAME}"  (My modem appears in /dev as ttyACM0)  newPPP.sh:  #!/bin/bash /usr/bin/pon prov $1 &gt;/dev/null 2&gt;&amp;1 &amp;   Problem:  The udev event fires, and newPPP.sh is running, but newPPP.sh process is killed after ~4-5s. ppp does not have time to connect (its timeout is 10s for dial up).  How can I run a long time process, that will not be killed?  I tried using nohup, but it didn't work either.  System: Arch Linux  Update  I found a solution , thanks to .  I use at now to run my job detached from udev process.  But the one question remains unanswered: Why do nohup and &amp; not work? How to filter string for unwanted characters using regex? Improved Help Center - site-specific pages and site-specific edits to all pages Right now, moderators can edit some pages in a site's Help Center. For example, the entire /help/on-topic page can be edited. However, the /help/dont-ask page can't be edited at all.  I'd like to see two changes to the Help Center:   All pages should be editable by site moderators. Some pages do apply to all sites, such as the /help/dont-ask page I mentioned above. However, it may be useful to edit in some site-specific guidance. The entire page doesn't need to be editable in order to ensure that key commonalities across the network remain in place on all sites. This uneditable content should be at the top of the page and moderators should be able to add site-specific content to the bottom. Moderators should be able to add new pages. Instead of maintaining meta tag [faq], FAQs can be added right into the Help Center from Meta. The Help Center serves as a single point of reference for the community rules, norms, and policies. Meta questions can be used to discuss updates to pages. This avoids issues such as question and answer ownership and the ability to accept/change accepted answers as users come and go from a site. An active site will always have diamond moderators to perform the edits to the Help Center.   Note that site-specific pages do already exist. , which no other sites have. This is something relevant to Stack Overflow (and perhaps a few other sites). It would be nice to allow sites that have this policy to link to their own Help Center for it and to edit the page to provide additional examples that may be relevant to their user base. How do I login to the Gamecenter on PewDiePie's Tuber Simulator, if you have not done it when they asked? How can I change the registered name in Windows 7 to my own? I am stuck at my exam practice here.      The remainder of the division of $x^3$ by $x^2-x+1$ is ..... and that of $x^{2007}$ by $x^2-x+1$ is .....   I tried the polynomial remainder theorem but I am not sure if I did it correctly.  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(In my mind, source current means the device can output a current with a positive voltage, while sinking current means the device takes in anything and connects it to GND, so perhaps that is where I am going wrong.)  Anyway I'm quite confused so any clarification would be much appreciated! Plotting recursive function using pgfplots \addplot Is it possible to plot a recursive function using pgfplots \addplot command?  I tried like this;  \documentclass{article} \usepackage{pgfplots}  \pgfmathdeclarefunction{fac}{1}{   \pgfmathparse{(#1&lt;=1)+                 (#1&gt;1)*#1*fac(#1-1)   } }  \begin{document} fac(5) is \pgfmathparse{fac(5)}\pgfmathresult  \end{document}   But the result seems to be an infinite recursion, as the entire function is being evaluated. Find files that a user can write, efficiently with minimal process creation I am root. I would like to know whether a non-root user has write access to some files - thousands of them. 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So I tried chsh -s /bin/zsh, but that doesn't work either. The Terminator  says:  How do I change my default shell? Terminator will always start the shell specified by the SHELL environment variable. (It will fall back to /bin/sh if the SHELL environment variable isn't set.) You could simply influence Terminator's environment, but the best solution is to tell your operating system which shell you'd like, and it will ensure that SHELL is set correctly for all applications, not just Terminator. See the documentation for your operating system for details on how to change your default shell.  So I expected my attempts to work, but they don't. How can I change the default shell in Terminator. Can all real polynomials be factored into quadratic and linear factors? Should I upgrade my body or lens first? $(X,\tau)$ needs to be $T_1$ in order to guarantee that $A'$ is closed? Claim: If $(X,\tau)$ is not $T_1$ then $A'$ is not necessarily closed for any $A$.  Proof: Here is an example. 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