Let S be a set of natural number that does not contain the digit 7 in their decimal representation.  prove that  $\sum_{n\in S}\frac{1}{n}$ &lt; ∞  help me please
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?
Permanently blocking a domain in iptables Suppose I want to block Facebook permanently. To do this, I have followed following processes:     #host -t a www.facebook.com   Sample output:     www.facebook.com has address 69.171.224.40   Find CIDR     #whois 69.171.224.40 | grep CIDR   Sample output:     CIDR:           69.171.224.0/19   To prevent outgoing access to www.facebook.com  Approach 1:     #iptables -I FORWARD -p tcp -d 69.171.224.0/19 -j DROP   Approach 2:     #iptables -I FORWARD -p tcp -d www.facebook.com -j DROP   Both Approaches work well. In approach 1, IP address may be changed so it will not block Facebook permanently. I don't know about approach 2 whether it will block Facebook permanently or not. If above approaches are not right way to block a domain permanently, how can I do it? I'm trying to find a copy of 6.20 to run a diff on a client's Drupal install. I believe there's been changes to core.   How can I get version 6.20? Additionally, how can I download any older Drupal versions? Why is pressure in a liquid the same in all directions? I'd like answers both in the more intuitive side an on the more precise side.   Thinking of water as "cubes" of water, for example, would allow pressure in the z axis to be independent of the y or x axis. Choosing other "shapes", like a  triangular prism, yield different results. Will this fact, then, be dependent on the format of the molecules of the liquid ?  I've heard of the rotational symmetry of liquids. What is its precise statement ? Why is it true ? How to set a command block with a command inside using the setblock command Does someone know how to set a command block with a command inside using the setblock command?  I have created this code /setblock &lt;x&gt; &lt;y&gt; &lt;z&gt; command_block {TileEntityData:{Command:"testfor @p[_=1]"}} but it doesn't works.  Do someone know how to do that? I am trying to analyze data based on a sample size of around 50 and would appreciate advice on which statistical test would be most appropriate. Participants, on a single questionnaire, were asked to complete two different sections. Variable 1: rating Likert items on a scale from 1-5 (strongly agree to disagree). Variable 2: rating Likert items of a validated tool on a scale of 1-9, which is averaged to obtain a single number.  My research question is: Do those who 'agree' with the statement in the first section significantly differ in terms of their scores on the variable 2, compared to those who 'disagree'? (ex. Do participants tend to 'agree' with "I like to read books for pleasure." have higher IQs (variable 2) than those who 'disagree'?)  I was advised by a colleague to run a one-way ANOVA with variable 1 (categorical) as IV and variable 2 (continuous) as DV.   Question 1: My independent variable is the three groups: 'agree', 'neutral', and 'disagree.' (I combined strongly agree with agree because of low frequencies and likewise for strongly disagree and disagree.) However, there are many more participants in the 'agree' category than in the 'disagree' category. The distribution is, on average, around 44 agrees, 3 neutrals, and 3 disagrees. Do the (very) unequal sample sizes prevent me from performing ANOVA, or any other tests? Which test would be most appropriate for this analysis?  Question 2: I would like to run analyses on all of the Likert items, which is more than 10 statements. (ex. Seeing how IQs correlate with agreement to many statements including “I like to read books for pleasure,” “I watch a lot of TV,” “I exercise more than 30 minutes each week,” etc.) I was planning to do a Bonferroni correction to adjust for the multiple tests. Would this be correct, and help make my analyses more robust? Note: The continuous data are not normally distributed (there is negative skewness), but I seem to be able to correct the skewness using logarithmic transformation.  Any advice or suggestions for papers to read would be appreciated. Thank you! Renting a car in Ireland & Great Britain: 1 rental or 2? My wife, two kids and I will be traveling in Ireland and the UK Nov. 1st through 21st. We're flying into and out of Dublin. Our plan is to do a clockwise tour: Fly into Dublin -> Belfast -> Scotland -> England/London -> Back to Dublin to fly home.  We've been planning to rent a car when we land in Dublin (doing just things outside Dublin at the start of the trip) and have a car all the way until London, then turn in the car and take a train/ferry to get back to Dublin before flying home.  However, reading through the car rental section of Rick Steve's 2012 Ireland book is causing us to question if this is the right plan.  He says (p. 480):     If your trip covers both Ireland and Great Britain, you're better off   with two separate car rentals, rather than paying for your car to ride   the ferry between the two islands. On an all-Ireland trip, you can   drive your rental car from the Republic of Ireland into Northern   Ireland, but be aware of drop-off charges ($75-150) if you return it   in the North.   So it seems like he recommends against doing one car rental for both islands, but he also recommends against renting in the Republic of Ireland and returning it in Northern Ireland (our other option for our current itinerary).  So we're not sure which is best here.  A few questions about this:   If we do one car rental all the way from Dublin to London, will there be a similar $75-$150 fee? Is there an easy place to return a car on the Northern Ireland side of one of the Northern Ireland -> Scotland ferries, and an easy place on the Scotland side to rent a car?  We'd like to minimize the number of buses/whatever we need to take between car rentals since we'll be lugging around car seats for both of our children (on top of carrying them and our luggage). Will one way be significantly cheaper than the other?  Besides the basic cost of taking the car on the ferry and the drop-off charges, our credit card will cover insurance in Scotland/England but not in Ireland, so we could potentially get a better rental rate by renting a second vehicle once we're on Great Britain. Although, I know you often get a better rate when keeping a car rental for longer, so that may negate any benefit from the credit card covering the insurance. Pointing the camera in a particular direction programmatically I'm trying to use Python to place a camera. I know the camera's location, and the forward and up vectors for the camera (i.e. the direction it's pointing and its orientation). How do I do this?  I know that it's possible to calculate a raw world matrix from the above information, which I can then apply to the camera, but I was rather hoping not to have to. Is there an easier way? How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$? (inverse of composition) I'm doing exercise on discrete mathematics and I'm stuck with question:     If $f:Y\to Z$ is an invertible function, and $g:X\to Y$ is an invertible function, then the inverse of the composition $(f \circ\ g)$ is given by $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$.   I've no idea how to prove this, please help me by give me some reference or hint to its solution. Applications Not Reopening After Restart Beginner here with an issue on a simple presentation using beamer  \documentclass{beamer}  \begin{document} \begin{frame}   \tau \epsilon \chi \end{frame} \end{document}   The above fails with the message "! Missing $ inserted." and waits for me to do something when converting to pdf. I have tried adding the instruction \usepackage[utf8]{inputenc} and also \usepackage[greek]{babel}. I have not seen references either to needing to use any other package. What is exactly I am missing? Way to list "Contact Us" email address on web site, yet reduce likelihood of spam? What's a good way to list a "Contact Us" email address on a web site, while reducing the likelihood it will get spammed?  Is putting the email address in an image the best technique, or are there others? How does a Windows antivirus hook into the file access process? How to make git ignore changes in case? I'm not too sure what is going on here, but sometimes a particular file in my repository will change the case of its name. e.g.,: before: File.h after: file.h I don't really care why this is happening, but this causes git to think it is a new file, and then I have to go and change the file name back. Can you just make git ignore case changes? [edit] I suspect it is Visual Studio doing something weird with that particular file, because it seems to happen most often when I open and save it after changes. I don't have any way to fix bugs in VS however, but git should be a bit more capable I hope. TikZ/PGF 3.0 datavisualization vs pgfplots To follow-up on a previous question , the current release of TikZ 3.0 has a flexible datavisualization engine that creates the same kind of plots as the more mature plotting package pgfplots.  What is the current recommendation for producing complete plots in LaTeX? Finite multiplicative group of matrices with sum of traces equal to zero. Let  $G=\{M_1,M_2,...,M_k\}$ be a finite set such that $ M_i\in M_n(\mathbb R)$ and $(G,\;\cdot\:)$ is a group with matrix multiplication. If $\sum _{i=1}^k \operatorname{tr} (M_i)=0$ then how to prove $\sum _{i=1}^k M_i=0$ ? Is there a way to get reverse numbering on the enumerate environment? To build a list of publications, I wish to show the most recent on top of the list, while keeping the numbering similar. This can be achieved manually by doing something like:  \begin{enumerate} \item[42] pub num 42 \item[41] pub num 41 \item[...] \item[1] pub num 1 \end{enumerate}   but is there a more principled approach? Why shouldn't we allow NULLs? Why does 64-bit Windows need a separate "Program Files (x86)" folder? Supremum and infimum of sets I've recently started to study supremum and infimum, and I'm having trouble determining the supremum and infimum of the following sets:  $ A = \left\{ {mn\over 1+ m+n} \mid m, n \in \mathbb N \right\} $ $ B = \left\{ {mn\over 4m^2+m+n^2} \mid m, n \in \mathbb N \right\} $ $ C = \left\{ {m\over \vert m\vert +n} \mid m \in \mathbb Z, n \in \mathbb N \right\} $ $ D = \left\{ {n\over 3} - \left[{n\over 3}\right] \mid m, n \in \mathbb N \right\} $   In $D$, the $[\; ]$ denotes the integral part of ${n\over 3}$.   $E = \{x \mid x \text{ is decimal fraction between } 0 \text{ and } 1 \text{ that has the digits } 0 \text{ and } 1\}$.  In my progress so far I was able to verify that sup $\{x \in A\} = +∞$  Setting $m = (n-1)$ to get  $$ \frac{mn}{1+m+n}= \frac{n(n-1)}{1 + n + n-1} = \frac{n(n-1)}{2n} = \frac{n-1}{2} $$  That way $\frac{n-1}{2}\in A$ for every $n\geq 2$. Thus, the supremum of $A$ is bigger than $\frac{n-1}{2}$ for every integer $n$.  I don't know how to go past this. x_x  Any help would be greatly appreciated! Let$\ p_n$ be the$\ n$-th prime. Can you give me a proof for$\ \prod_{i=1}^\infty \frac{p_i-1}{p_i}=P\approx \frac{1}{11.0453}$? I found$\ \prod_{i=1}^\infty \frac{p_i}{p_i-1}\approx 11.0453$ on . Moreover, writing a paper, should one provide a proof or it is trivial? Thanks in advance.