Here's Prob. 7 in the Exercises of Chapter 3 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:      Prove that the convergence of $\sum a_n$ implies the convergence of $$ \sum \frac{\sqrt{a_n}}{n}$$ if $a_n \geq 0$.    How to show this? I have no clue! Can anybody here please be of any help?  Is this result a special case (or application) of a more general result about infinite series?
Ch.3 #7. Prove that the convergence of $\sum a_n$ implies the convergence of $$\sum \frac{\sqrt{a_n}}{n},$$ if $a_n \geq 0$.      My attempt. If $\sum a_n$ is convergent, then by the root test, $\lim_{n \rightarrow \infty} sup \sqrt[n]{|a_n|} &lt; 1$. This means $|a_n|^{1/n} &lt; 1$. Because $a_n \geq 0$, this means $a_n ^{1/n} &lt; 1$. Now apply the root test to $\sum \frac{\sqrt{a_n}}{n}$ by considering $$\lim_{n \to \infty} sup \sqrt[n]{|\frac{\sqrt{a_n}}{n}|}.$$ $$\sqrt[n]{|\frac{\sqrt{a_n}}{n}|} = \frac{\sqrt{{a_n}^{\frac{1}{n}}}}{\sqrt[n]n}.$$ Because  $a_n ^{1/n} &lt; 1$, certainly the numerator on the left-hand side of the equation above is $&lt;1$. And because $n$ tends to $\infty$, and $n &gt; 1 \rightarrow \sqrt[n]n &gt; 1,$ the denominator is $&gt;1$. Therefore the limit is $&lt;1$ and the root test implies that $\sum \frac{\sqrt{a_n}}{n}$ converges.     If there are any errors in my proof, I think they come from 1) my assumption of the converse of the root test theorem and 2) my misunderstanding of $\lim_{n\to\infty} sup$. Thanks in advance for your help.
Undefined control sequence \documentclass First of all I would like to thank you because you helped me out a lot with all your answers to the already online questions.  Now, here is my problem. I am trying to do a bibliography file with TeXworks but I get this error when I try to compile (pdfLaTeX+MakeIndex+bibTeX):      This is pdfTeX, Version 3.1415926-2.4-1.40.13 (MiKTeX 2.9 64-bit)     entering extended mode     ****PATH here******     ! Undefined control sequence.     l.1 \documentclass         [a4paper,11pt]{report}   I've been reading some similar questions but I don't find the way to fix it (maybe I am missing something). I found one that says that I am trying to compile a LaTeX file using TeX compiler but I don't know how to change the configuration.  This is the .bib fiel content:      \documentclass[a4paper,11pt]{report}     \usepackage[english]{babel}     % faire de l'anglais     \usepackage[latin1]{inputenc}   % accents dans le source     \usepackage[T1]{fontenc}        % accents dans le DVI     \usepackage{url}     \begin{document}      @Article{ref11, author={Danihelka, Jiri and Kencl, Lukas}, journal={Cloud Futures Workshop 2012 }, title={Colaborative 3D Environments over Windows Azure}, year={2012}, month={May}, pages={}, note={}     }      @book{ref12, author = {Lee, Henry and Chuvyrov, Eugene},     title = {Beginning Windows Phone 7 Development},     publisher = {Apress},     volume = {},     number = {},     series = {},     address = {},     edition = {Second edition},     year = {2011},     month = {July},     note = {}     }      @Article{ref13, author={}, journal={}, title={Cloud computing}, year={}, month={}, pages={}, url={\url{http://en.wikipedia.org/wiki/Cloud_computing}}     }      \end{document} Let $x, y$ and $z$ be real numbers such that $x^2+ y^2+ z^2= 1$. Find the   maximum and minimum values of $2x + 3y + z$ .       How can I able to solve the problem? Thanks for your help. I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at  and I understand the proof, but I don't "get it". Can someone help me out with this ?   I find it hard to wrap my head around the idea of how the column space and the row space is related at a fundamental level. trouble installing pygame on ubuntu 16.04 LTS I had a hard time installing pygame on my ubuntu 16.04 LTS. I use python 3. these are the steps I took to install pygame:  1.first I installed pip:  sudo python3 get-pip.py   this is the output of running pip --version:  pip 10.0.1 from /usr/local/lib/python3.5/dist-packages/pip (python 3.5)    I executed the following commands:   sudo apt-get install python3-dev mercurial  sudo apt-get install libsdl-image1.2-dev libsdl2-dev libsdl-ttf2.0-dev   then I executed the following command:   pip install --user hg+http://bitbucket.org/pygame/pygame  and this is the result in terminal:  Collecting hg+http://bitbucket.org/pygame/pygame   Cloning hg http://bitbucket.org/pygame/pygame to /tmp/pip-req-build-hxy86__i abort: error: Connection reset by peer Command "hg clone --noupdate -q http://bitbucket.org/pygame/pygame /tmp/pip-req-build-hxy86__i" failed with error code 255 in None   and it fails to install pygame. what should I do?? How to use XMLHttpRequest to download an HTML page in the background and extract a text element from it? Let $T$ be a linear transformation $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $T^{2}=\lambda T$ for some $\lambda\in\mathbb{R}.$ Then which of the following is/are true?  $1.\|T(x)\|=|\lambda|\|x\|$ for all $x\in\mathbb{R}^n.$  $2.$ If $\|T(x)\|=\|x\|$ for some non-zero $x\in\mathbb{R}^{n}$, then $\lambda=\pm 1.$  $3.T=\lambda I$ where $I$ is the identity transformation on $\mathbb{R}^{n}.$  $4.$ If $\|T(x)\|&gt;\|x\|$ for a nonzero vector $x\in\mathbb{R}^{n},$ then $T$ is singular.  According to me since $T^{2}=\lambda T$ we have $T$ satisfies the polynomial $x^{2}-\lambda x.$ So choices for its minimal polynomial are $x,x-\lambda,x(x-\lambda).$ So $3$rd option is clearly incorrect and $4$th option is not satisfied if $T=I$ and hence correct one. I am confused in first two options . Please help me. Thanks in advance. Display lost/negative reputation in topbar achievements When you get reputation, you get a +X green spot on your reputation feed telling you that you gained reputation and if you drop down the menu, you get more detail.  How about we make it that when you lose more than you got since the last time you checked, you see it red with a little -X.  For example, let's say you've just looked at the reputation bar so any old rep notification is gone and you ask a new question. 3 users downvote your question and 1 upvotes it. You will see it green because you gained 5 rep but you should see it red because you are at -1 rep.   This would help people improve their questions / answer by seeing, as soon as they get on the site, that they got negative feedback and that something needs to be checked / changed.    It shows here so it could very well be applied to the notification count: This actually triggered me in my mind from . After some playing around I notice that the relation $a^{\log_b x} = x^{\log_b a}$ is true for any valid value of $a,b$ and $x$. I am very inquisitive to see how this holds ? Here is a question as phrased in Advanced Linear Algebra (Steven Roman):     Show that a linear operator $\tau$ on a finite-dimensional complex inner product space $V$ is normal if and only if whenever $S$ is an invariant subspace under $\tau$, so is $S^\perp$.   I'm only interested in the $\Rightarrow$ direction. Wikipedia (and ProofWiki)  that uses some kind of computation with the trace. Linear Algebra Done Right chooses an orthonormal basis and then uses the fact that $\Vert \tau v \Vert = \Vert \tau^* v \Vert$.  However, the above  proofs seem a bit too computational to me. Can the following argument work?  (Use $\odot$ for orthogonal direct sum.) Suppose that $\tau$ is normal and write $$V=E_{\lambda_1} \odot \cdots \odot E_{\lambda_k}$$ where the $E_{\lambda_i}$ are the eigenspaces of $\tau$. If $S$ is a $\tau$-invariant subspace then $\tau|_S$ is diagonalizable, so $$S=\widetilde{E}_{\lambda_1} \odot\cdots\odot \widetilde{E}_{\lambda_k}$$ where the $\widetilde{E}_{\lambda_i} \le E_{\lambda_i}$ are the eigenspaces of $\tau|_S$ (which could be trivial). Then choose orthogonal complements $D_i$ such that $E_{\lambda_i} = \widetilde{E}_{\lambda_i} \odot D_i$, and then $$S^\perp = D_1 \odot\cdots\odot D_k.$$ Since each $D_i$ is a subspace of an eigenspace, they are all $\tau$-invariant. Clipping image with Basic (ArcView) level license of ArcGIS Desktop? Is there a way to trim a geo-referenced air photo to an irregularly shaped boundary?    I tried the clip tool shown in my attachment, but it didn't work.  It looks as though the output extent field requires a rectangularly shaped polygon as the clipping feature.    Can I even do this in ArcView?    I've never had a request to do anything like this before.        I do use Photoshop and other similar software packages.  I could just go in to an environment like this &amp; start painting everything outside of my boundary in white, but then I would have to come back in and geo-reference the image again, and it just strikes me that there must be a faster and easier way (even though I only have ArcView 9.3). How to read such figures with % as "up (by) 120%" and "increased (by) 120%". Do they mean the same? Selecting text in profile activity comments causes unexpected clipping I don't know what I've done, but all of a sudden I am unable to compile any document which uses the geometry package. Minimal (non)-working example:     \documentclass[11pt]{article}      \usepackage{geometry}  \begin{document} test \end{document}   doesn't produce any output, and this is what I see in my error summary screen in TeXnicCenter:    The red arrow points to what seems to be the problem: something about the bundle "oberdiek" and its package "ifvtex" is causing pdflatex.exe (which, while I don't really know what it does, sounds pretty important) to "not be initialized". I , and it's relevance here is a mystery to me.  I've confirmed that all of my files without the geometry package compile just fine; however, that doesn't mean that the problem is with the geometry package - what seems more likely is that there is some hidden cause of the error, and geometry happens to be one way to induce that. By the way, I've tried compiling a document which includes the ifpdf and ifvtex packages separately, and they do just fine; it's only when I include the geometry package (which for some reason requires these ifpdf and ifvtex packages) that the error occurs.  I've tried the only things I can think of:   going into MiKTeX Package Manager (by the way I am using 2.8), and removing + reinstalling both geometry and oberdiek refreshing the file name database ("FNDB") deleting all automatically generated files, as well as simply starting over in a new document   So I'm stuck now. I've considered attempting to just get rid of all traces of MiKTeX and TeXnicCenter from my computer and starting over, but that seems a bit excessive - after all, the geometry package was working for me . I honestly am not sure what I did to cause this; I've spent today trying to make a better template file for myself, and so I've been trying out new packages, but if I'm not loading these new packages there shouldn't be any conflicts, right?   Actually, I've just now thought of something that I think I did between the last time it worked and now, so is a possible candidate for the cause: I had to install some package manually, as it wasn't available on MiKTeX, and so I created a folder and pointed MiKTeX to it (i.e. the second option ). The package didn't do what I wanted, so I deleted the folder and removed it from the list of folders I told MiKTeX to check. Perhaps it's still trying to reach it for some reason? If $\lambda$ is an eigenvalue of $A^2$, then either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A$ $A$ is an $n\times n$ matrix of complex numbers. Prove that if $\lambda$ is an eigenvalue of $A^2,$ then $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A.$    If $\lambda$ is an eigenvalue of $A^2,$ we have $\lambda\alpha=A^2\alpha$ for some $\alpha.$ Then how can we find a $\beta$ s.t. $\sqrt{\lambda}\beta=A\beta?$ Why is integration so much harder than differentiation? It would be nice to be able to put emphasis on specific bits of a fragment or of a quote (bold can help for a quote but I find that it makes things less readable sometimes). What do you think? Update: On , you can highlight particular lines by prefixing each line with @@. See also: Missing "References" heading How to Tail -f output with colors using only Awk and show the rest of the output? I was able to do a tail -f output with colors using the awk command. But I'm unsure how to output the rest of the log as it seems to be filtering only those 2 lines. I also want to see the other lines as the log is written.  What do I need to put in the awk command to show the rest of the output?  Image of my issue. The red text above is without using awk and displays the entire tail -f log. And below that is the same command with the awk color filtering. Preserve search engine rankings while shifting to new domain name A certain physics problem I have been working on has turned into a math problem. Particularly, I want to find the solutions of some equation of the form   $$f(x)=ae^{bx}+cx+d = 0$$  where $a, b, c,$ and $d$ are constant, real numbers that come from the physics problem and $x$ will be a real number. I do not know how to find the solutions of an equation of this form. In the special case where $c=0$, the solution is simply  $$x = \frac{\ln(\frac{-d}{a})}{b}$$  but for other values of $c$, I am stumped. If there is no exact form for the roots, is there a relatively simple way to estimate the roots?