Divergence of series - Counter example
Prove or disprove: if $\sum\limits_{n=0}^∞a_n$ converges and $\lim\limits_{n\to\infty}b_n =0$, then $\sum\limits_{n=0}^{\infty}a_n b_n$ converges.
A question in Complex Analysis $\int_0^{2\pi}\log(1-2r\cos x +r^2)\,dx$ My problem is to integrate this expression:  $$\int_0^{2\pi}\log(1-2r\cos x +r^2)dx.$$ where $r$ is any constant in $[0,1]$.  I know the answer is zero. Can you explain you idea to me or just prove that? Maybe you will use the "Cauchy integral theorem ". How to update kernel to the latest mainline version without any Distro-upgrade? Currently I'm running Linux-3.0 and I want to update it to Linux-3.3.1 the latest stable kernel release as mentioned at . Can I update to 3.3.1 in Ubuntu without any risk of crashes? I'm updating my kernel regularly as provided by the Update Manager..Currently I have Linux-3.0.0.17..Can I update?  NOTE:  The latest kernel version any user is supposed to use in Ubuntu is updated automatically via the Update Manager, so no action is normally required by user regarding kernel upgrades. What the question author is referring to is mainline kernel, see: There is a socket related function call in my code, that function is from another module thus out of my control, the problem is that it blocks for hours occasionally, which is totally unacceptable, How can I limit the function execution time from my code? I guess the solution must utilize another thread. Can a self-complementary graph on 100 vertices have precisely one vertex of degree 50? How to draw arrows between circled coefficients of polynomials using tikz After reading Gonzalo's answer to , I thought  I would be able to modify it so that I could have the corresponding coefficients circled and an arrow going from one circle to the next. For example, I would like the coefficient of x^2 on the left, a, circled and the coefficient of x^2 on the right, 5 circled, then a curved arrow drawn between the two circles. How might I achieve this? Also, how would I add some text along the path of the arrow? I reproduce his code here.  \documentclass{article} \usepackage{amsmath} \usepackage{tikz} \usetikzlibrary{calc}  \newcommand{\tikzmark}[1]{\tikz[overlay,remember picture] \node (#1) {};}  \begin{document}  \begin{equation}   a\tikzmark{a}x^2 + bx + c = 5\tikzmark{b}x^2 + bx + c.   \begin{tikzpicture}[overlay,remember picture,out=315,in=225,distance=0.4cm]     \draw[-&gt;,red,shorten &gt;=3pt,shorten &lt;=3pt] (a.center) to (b.center);   \end{tikzpicture} \end{equation}  \end{document} Scanner is skipping nextLine() after using next() or nextFoo()? If the brain has no pain receptors, how come you can get a headache? I've read many years ago in books, that the brain has no nerves on it, and if someone was touching your brain, you couldn't feel a thing.  Just two days before now, I had a very bad migraine, due to a cold. It's become better now, but when I had it I felt my head was going to literally split in half, as the pain was literally coming from my brain.  So it lead me to the question: How come people can get headaches if the brain has no nerves? I am trying to pass the email value from a data extension to HTML. I am using the CloudPagesURL() in my email in attempt to grab the subscriber's email.  Here is my code syntax that I am attempting to use:  Email Code:  &lt;p&gt;&lt;a title="MyPage" href="%%=RedirectTo(CloudPagesURL(505,'Email Address',@EmailAddress))=%%" alias="ThisAlias" conversion="false"&gt;Click to unsubscribe page 505&lt;/a&gt;&lt;/p&gt;   CloudPage AMPScript Code:  %%[  VAR @emailAddress  SET @emailAddress = RequestParameter("emailaddr")  ]%%   Cloudpage HTML Code:  &lt;td width='155' valign='bottom'&gt;&lt;input type='text' name='Name1' value='%%=v(@emailAddress)=%%' id='Name1' /&gt;&lt;/td&gt;   Any reason as to why this doesn't work? My syntax might just be wrong or I might be using the CloudPagesURL() function incorrectly. This integral is very closely related to the sophmores dream that states  $$ \int_0^1 x^{-x}\mathrm{d}x = \sum_{n=1}^\infty n^{-n} = 1.27\ldots $$  For example here   Now I want to bound the integral, and showing that is less that 2.  For the interval $[0,1]$ a good bound is rewriting it to $\exp(x\log x)$ and using the expansion   $$ 1 - x \log(x) + \frac12 (-x \log(x))^2$$  but how does one handle $[1,\infty)$ ? In this answer here     gives bounds to the integral, but they are not tight enough.. So to taste my question again, how does one prove that   $$ \int_0^\infty \frac{\mathrm{d}x}{x^x} \leq 2 $$ What was missing in Dirac's argument to come up with the modern interpretation of the positron? How to properly render smoke on a transparent background? When rendering volumes with Film > Transparent enabled, the alpha appears to be the wrong type (premultiplied/straight). In the 3D view it works as expected:    This is a known .  Is there any way to work around this? How do I install Ubuntu on a VirtualBox client from an ISO image? I have the ISO image file (that is, *.iso) and I am trying to boot my  installation with this image. Is it possible? I know how to mount my ISO image to my filesystem (on my host machine), but I don't think VirtualBox boots an OS from a location on my file system.  I've even tried to mimic mounting, such that VirtualBox thinks it has to boot from the CD drive, but this failed.   What is the correct way to boot an OS from an ISO image in VirtualBox?  My host machine is Ubuntu 10.04 (Lucid Lynx) and my VirtualBox version is 4.1.4. Upload images from the Stack Exchange Android app I would like to request implementation of uploading images to questions/answers from the Stack Exchange Android app, especially now as it would definitely help uploading screenshots for reporting bugs and feature requests. On a quantum scale the smallest unit is the , which is a discrete measure.  There several question that come to mind:   Does that mean that particles can only live in a discrete grid-like structure,  i.e. have to "magically" jump from one pocket to the next? But where are they in between? Does that even give rise to the old paradox that movement as such is impossible (e.g. )? Does the same hold true  for time (i.e. that it is discrete) - with all the ensuing paradoxes? Mathematically does it mean that you have to use difference equations instead of differential equations? (And sums instead of integrals?) From the point of view of the space metric do you have to use a discrete metric (e.g. the ) instead of good old Pythagoras?   Thank you for giving me some answers and/or references where I can turn to.  Update: I just saw this call for papers - it seems to be quite a topic after all: Is Reality Digital or Analog? FQXi Essay Contest, 2011. , , . One can find some pretty amazing papers over there. 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It seems that something at least remotely easy could be done for batch files in Windows. I'm looking for a solution that works in a standard Windows command prompt, without any extra utilities. Please no PowerShell or Cygwin either. Cannot find Background Images feature to add in a photo in Blender 2.8  Where is it?