Sum of the inverse of all positive integers which do not contain the digit 8 Let $No8$ be the set of positive integers that do not contain the digit $8$. For example, $123456790 ∈ No8$ but $1234567890 \notin No8$. Show that $$\sum_{n\in No8} \frac 1n&lt;80$$  The bound in the above inequality is not the best possible; what is the best upper bound for $\sum_{n\in No8} \frac 1n$?
Does the harmonic series converge if you throw out the terms containing a $9$? I found this very amusing comic on the internet the other day:   The last frame seems to claim that the harmonic series converges if you throw out all the terms with a $9$ in the denominator. Is this true? Where could I find a proof? Is it only true for terms with a $9$ or is it true for any number? If it's only true for $9$, why is that?
How can I compute $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})$? How can I compute $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})$?  If $\varphi$ is one of this automorphism then $\varphi((1,0))=(1,0),(1,3),(3,3),(3,0)$ and $\varphi((0,1))=(0,1),(0,5),(2,1),(2,5)$.  So it seems to me that this groups contains 16 elements, am i right? And if I'm right there is a way to understand to which group of cardinality 16 this is isomorphic to?  EDIT: $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}\cong\mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$. And $\mathbb{Z}/8\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$ are characteristic (am I right?), so $\mathrm{Aut}(\mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z})\cong\mathrm{Aut}(\mathbb{Z}/8\mathbb{Z})\times\mathrm{Aut}(\mathbb{Z}/3\mathbb{Z})\cong\mathbb{Z}/8\mathbb{Z}^*\times\mathbb{Z}/3\mathbb{Z}^*$ and this group has cardinality 8, right? Where is my mistake? How can I bind the meta key (a.k.a.  / Windows key) to a left click? Hitting the meta key should trigger a left click, and pressing down on it should allow me to drag and drop (just like it would do when holding down left click).  I tried using System Preferences -> Shortcuts and Gestures:    but I cannot set "meta" alone as being a shortcut.  I use Kubuntu 14.10 Plasma 4 and Kubuntu 14.04 LTS x64.   (On Windows 7 SP1 x64 Ultimate I can achieve that using the program .) How to prove this sequent using natural deduction? Is there a way to find the number of files of a specific type without having to loop through all results inn a Directory.GetFiles() or similar method? I am looking for something like this:  int ComponentCount = MagicFindFileCount(@"c:\windows\system32", "*.dll");   I know that I can make a recursive function to call Directory.GetFiles , but it would be much cleaner if I could do this without all the iterating.  EDIT: If it is not possible to do this without recursing and iterating yourself, what would be the best way to do it? Creating a dictionary from a CSV file PDOException “could not find driver” Unable to see anything after importing .obj file WP_query 'orderby=none' Problem I have another database which stores post's IDs and I want do display them using WP_query to use the WP default pagination. I'm using orderby => none to preserve the original order, but the result reorders them by IDs.  $ids = array (60, 23, 78, 46, 105, 130)  $args = array (     'posts_per_page' =&gt; -1,     'post__in' =&gt; $ids,     'orderby' =&gt; 'none' );  $query = new WP_Query ( $args );  while ( have_posts() ) : the_post();     echo '&lt;li&gt;';     the_title();     echo '&lt;/li&gt;'; endwhile;   Could anyone help me?  Thanks, Romulo De Lazzari Why does TeX Live "require" yearly updates? Every year TeX Live provides a new version. My recollection is that soon after a new version is released, the old version's tlmgr stops fetching updates to packages. They provide  with the disclaimer     This procedure is not bullet-proof, or especially recommended   There is no upgrade path for Windows. Looking at the , it is not clear to me what is actually changing between releases such that a full new install is required/recommended. I always thought of TeX Live as providing useful binaries (e.g., pdflatex, biber, and makeindex) and a convenient way of updating  packages. What am I missing? Why does TeX Live need a new install every year? What is the proper way to explain the twin paradox? Using OPENQUERY to execute a script I have a SQL Server 2008 instance which I want to use to import data from an Oracle server.  I have set up a linked server that works correctly when running simple queries like SELECT * FROM table. However, if I declare a variable or loop through the rows in a table or anything else inside the OPENQUERY, I get errors.  Is it possible to do this using OPENQUERY? Do I need additional permissions? AirPort Time Capsule wifi is suddenly very slow: Boom network load Changing default \citet font in biblatex I'm using biblatex to manage my citation in a natbib style :  \usepackage[style=alphabetic,maxnames=6,natbib=true]{biblatex}   However when I use the \citet command, the author's name is in a \textsc font, I would like to change this behavior so that the font used is the default one, but I could not find how to do this in biblatex's documentation. How do I install pillow in Ubuntu 16.04 LTS? I went to  for installing Pillow since the original PIL is a discontinued project and so now not so good to use. But, in the table support for Ubuntu 16.04 LTS is unmentioned. It appears neither in , or in .     Further, just installing pillow is said to be not enough. There are some dependencies that should be installed first which is also unmentioned for Ubuntu 16.04 LTS. Would the installation steps for Pillow for python 3 given for 14.04 LTS be OK for Ubuntu 16.04 LTS?  If not, How do I install pillow in 16.04 LTS for Python 3.6? (There is support for Python 2.) What is the difference between SAN, NAS and DAS? What is the difference between SAN, NAS and DAS? I am trying to add smoke to a cigarette. I've created a Cylinder, hit space and chose quick smoke while this sphere was selected. It created a transparent bigger sphere and it is emitting smoke in solid and texture mode. But when I try it out in render mode, it won't show the smoke for some reason. Here is my setup:    The cylinder has a basic white material. Thanks! How can I translate CMYK to Pantone colors? In particular, how do I translate CMYK 90C 85M 80Y 63K to a Pantone color? Is this a valid partial fraction decomposition? Write $\dfrac{4x+1}{x^2 - x - 2}$ using partial fractions. $$ \frac{4x+1}{x^2 - x - 2} = \frac{4x+1}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x-2} = \frac{A(x-2)+B(x+1)}{(x+1)(x-2)}$$ $$4x+1 = A(x-2)+B(x+1)$$ $$x=2 \Rightarrow 4 \cdot2 + 1 = A(0) + B(3) \Rightarrow B = 3$$ $$x = -1 \Rightarrow 4(-1) +1 = A(-3)+ B(0) \Rightarrow A = 1$$ Thus, $$\frac{4x+1}{x^2-x-2} = \frac{1}{x+1} + \frac{3}{x-2}\textrm{.}$$  The substitution of $x$ ($x = 2, -1$) is a common method to find out the coefficient of the partial fractions. However, the equation on the third line is obtained by multiplying $(x+1)(x-2)$, which is assumed to be nonzero. Here we have a contradiction. Furthermore, the original function is not defined at $x=-1,2$. How can we substitute these value for $x$? So is this method valid and rigorous? How to modify it so that it is rigorous? I heard that there was a twitter bot which automatically posted questions and asked for answers for specific questions on every site. How do they work? Is there an algorithm for doing this or is it purely random? I know that it doesn't tweet closed questions, but that's it. Submitting ACM Conference papers to ArXiv I'm a little confused. The  states:     ACM authors hold the right to post copies of their own peer-reviewed accepted ACM works on any non-commercial repository or aggregation that does not duplicate ACM tables of contents, i.e., whose patterns of links do not substantially duplicate an ACM-copyrighted volume or issue. Non-commercial repositories are here understood as repositories owned by non-profit organizations that do not charge a fee for accessing deposited articles and that do not sell advertising or otherwise profit from serving articles.   However, the  site states:     Can I publish the published version of my publication on arXiv?       No. An ACM author cannot grant arXiv.org or any other publisher rights to distribute their works that have been published by ACM.   SHERPA/Romeo shows ACM as being a "Green Publisher", which would support the case that I can upload to ArXiv.   states that I can upload pre- or postprints under certain circumstances, but does not explicitly mention Arxiv, positively or negatively.  So, what is it? Can I submit a paper I published with an ACM Conference (ACM WiSec, if that makes a difference) to ArXiv or not? As I understand it, ArXiv is a non-profit, so it would fit the terms of the publishing policy. Or is this related to the rights you have to grant ArXiv upon submission?  If I can only publish a specific version of the paper, which one?