If I offered to give you \$10 but steal $5 from you, you should take that deal. You should take it once, and again, and again. You will always gain money no matter how many times you take it and the ratio of money gained to money lost is constant. But, what if you took the deal an infinite number of times? You will have both gained and lost an infinite amount of money. It's not as if you can have gained 10∞ and lost 5∞. The infinities have to cancel out, right? But, then, at each step, the ratio of money gained to money lost is unchanged but, after an infinite amount of time, it is changed. That's clearly impossible, so what actually happens then? What am I thinking about wrong here?
A friend of mine asked me the following question:      Suppose you have a basket in which there is a coin. The coin is marked   with the number one. At noon less one minute, someone takes the coin   number one and put into the basket two coins: the number two and the   number three. At noon less $\frac{1}{2}$ minute, he takes the coin   number two and put into the basket the coins number $4$ and the number   $5$ and so on. At noon less $\frac{1}{k}$ he takes the coin number $k$   and put two other coins.       The question is: How many coins I can find in the basket at noon?   Intuitively the answer is $\infty$ because I added one coin for infinite times. But the correct answer seems to be $0$ because every coin numbered $k$ has been removed at noon less $\frac{1}{k}$ of minute. What is the correct answer? Thanks.
I have to show by induction that this function is a multiple of 8. I have tried everything but I can only show that is multiple of 4, some hints? The function is $$5^{n+1}+2\cdot 3^n+1 \hspace{1cm}\forall n\ge 0$$, because it is a multiple of 8, you can say that$$5^{n+1}+2\cdot 3^n+1=8\cdot m \hspace{1cm}\forall m\in\mathbb{N}$$. Efficiently selecting a set of random elements from a linked list From :   (formal) showing great care and effort in your work synonym: diligent  (formal) working very hard and taking great care that everything is done as well as it can be synonym: diligent  (formal) showing care and effort in your work or duties  These definitions look alike. What they have in common is that these words mean to take care and to put effort into your work. All words are formal. None of these seems to be a proper term in a specific field, e.g. medical, law. I also looked at  to get more details. This adds the connotation of steady and earnest to diligent and unremitting attention and persistent application to assiduous. But in the end, these additional information are again describing an equal thing, i.e., continuously doing your work. The entry on sedulous doesn't add any more content; it refers to diligent and again describes the word with perseverance. From this point (checking dictionaries only), I don't see any difference at all. I continued with a corpora search. From : Diligent is most used in combination with work, effort, research, study but also student, people.  She's a diligent student. His diligent research identified the value of ...  Assiduous is also most used with work, effort, research but not with student, people. Merriam-Webster's Learner's Dictionary entry on  lists student as example though.  He's assiduous in his work. With assiduous attention to detail ...  Sedulous is most used, um, to little results.  has even less content. Merriam-Webster's example is, once again, a student. So, diligent is used most often, followed by assiduous and sedulous is quite rare. All words are formal. All words seem to be usable as an attribute to a person (e.g., student) but also specifying the intense care and effort on work, e.g., a research. I don't see any significant difference in these words, besides frequency. When is a student diligent, assiduous or sedulous? And when is a research diligent, assiduous or sedulous? Or simply: What are the subtle distinctions which aren't represented by dictionaries? Getting my File Manager to link to my Google Drive I have a very small amount of storage on my computer and would like to conserve it, so I would like to know if there is a way to mount my Google Drive in Thunar or Nautilus on Ubuntu 12.04.5 LTS.   Yes, I know, it's weird that I'm saying that I have a small amount of storage when I have 2 file managers installed, but I literally have 5GB out of 16GB left. There is a remote Windows server on a private network which I can connect to via Remote Desktop Connection.  I would like to be able to make TCP/IP connections from my computer to other computers on that server's network.  Remote Desktop Connection makes it possible to share printers, drives, and other local resources through the connection.  Is there any way to "tunnel" a TCP/IP connection via RDC?  I'd like something similar to the port-forwarding provided by SSH.  I don't see any way to do this via RDC, but I'm hoping the capability is there and I just don't know about it. I have 3 fields ( Exposed filter ) 1 Country 2 State 3 City  When a respective Country is selected in the 1st field, I want the second filter ( Dropdown or autocompete ) to display states of the country selected in the 1st field &amp; when this is done based on the state selection , I want the 3rd field to show only the cities belonging to the state selected in the second field.  How can this be done ? any help ? When selecting components, is there a way to control the "selection distance/area"? I don't want to select a component, even if I click ~20 pixels away from it.    Thanks in advance! I am currently using 19.04 and I got "No new release found for 19.10" Aliens like giant grasshoppers invade earth, give black people power over whites Biber 2.11: Data file ... cannot be read in encoding UTF-8 ... ill-formed UTF-8 octet sequence ... at ... Slurper.pm line 63 I have just changed computers, and running Biber 2.11 now produces this error:  Utils.pm:209&gt; ERROR - Data file ... cannot be read in encoding 'UTF-8':  Can't decode ill-formed UTF-8 octet sequence &lt;96&gt; at C:\ ... Slurper.pm line 63   My .bib file is huge, having accumulated over years.  Thus, I suspect that there's something malformed in it.  Is there any rapid way to trouble-shoot it?  The new machine runs Win 10, Biber 2.11, XeLaTeX 2018.10.31, LaTeX2e 2018-04-01 and TeXLive 2018.  The old machine ran Win 7 and slightly earlier versions of all of the above.  A MWE of the driver .tex file:  \documentclass{article} % also with beamer  \usepackage[utf8]{inputenc} % problem arises whether or not include this \usepackage[backend=biber,bibencoding=utf8]{biblatex} %  problem arises whether or not include this \addbibresource{bib-mwe.bib}  \begin{document}  cite reference \cite{mal-rwws}  \printbibliography  \end{document}    Thanks! TRON: Game A.I algorithm? Correlation between dichotomous and continuous variable I am trying to find the correlation between a dichotomous and a continuous variable.  From my ground work on this I found that I have to use independent t-test and the precondition for it is that the distribution of the variable has to be normal.  I performed Kolmogorov-Smirnov test for testing the normality and found that the continuous variable is non-normal and is skewed (for about 4,000 data points).   I did the Kolmogorov-Smirnov test for the entire range of variables. Should I split them into groups and do the test? I.e., say if I have risk level (0 = Not risky, 1 = Risky) and cholesterol levels, then should I:    Divide them into two groups, like  Risk level =0 (Cholestrol level) -&gt; Apply KS Risk level =1 (Cholestrol level) -&gt; Apply KS  Take them together and apply the test? (I performed it on the whole dataset only.)   After that, what test should I do if it is still non-normal?    EDIT: The above scenario was just a description I tried to provide for my problem. I have a dataset which contains more than 1000 variables and about 4000 samples. They are either continuous or categorical in nature. My task is to predict a dichotomous variable based on these variables (maybe come up with a logistic regression model). So I thought the initial investigation would involve finding the correlation between dichotomous and a continuous variable.  I was trying to see how the distribution of the variables are and hence tried to go to t-test. Here I found the normality as an issue. The  Kolmogorov-Smirnov test gave a significance value of 0.00 in most of these variables.  Should I assume normality here? The skewness and kurtosis of these variables also show that the data is skewed (>0) in almost all cases.  As per the note given below I will investigate the point-biserial correlation further. But about the distribution of variables I am still unsure. Prove $\sum{\frac{1}{(x+2y)^2}} \geq\frac{1}{xy+yz+zx}$ Is Canon 60D still a decent camera? I have a Canon 60D, bought it back in 2013. Use it quite a while, but in past two-three years, I lost interest in photography. I will be going on a long trip this summer and would like to take a decent camera with me. Is Canon 60D still a decent one or I will be missing on features and upgrades comparing to the latest versions?  In terms of lenses I own Tamron 24-70 2.8 And Canon 100 2.8 Macro. How to stop a sponge cake (chiffon cake) from deflating/sinking? Biblatex - capital names in references, regular in cites I am trying to achieve a biblatex style that would do the following:  in the case of \parencite{} I would get ...like that (Smith 2010).   in the case of \textcite{} I would get Smith (2010) says that...  but in References I would get SMITH, J. Bookname.  How can I achieve this?  I use \usepackage[style=authoryear, firstinits=true]{biblatex}  I tried \renewcommand*{\mkbibnamelast}[1]{\MakeUppercase{#1}} but that makes the name uppercase even in the text, which is undesirable especially with \textcite{} I recently came across the paper . The author raised a few common misconceptions regarding hypothesis testing and p-values, about which I have two specific questions:   The p-value is technically $P({\rm observation}|H_{0})$, which, as pointed out by the paper, generally does not tell us anything about $P(H_{0}|{\rm observation})$, unless we happen to know the marginal distributions, which is rarely the case in "everyday" hypothesis testing. When we obtain a small p-value and "reject the null hypothesis," what exactly is the probabilistic statement that we are making, since we cannot say anything about $P(H_{0}|{\rm observation})$? The second question relates to a particular statement from page 6(652) of the paper:      Since the p-value, or range of p-values indicated by stars, is not set a priori, it is not the long-run probability of making a Type I error but is typically treated as such.   Can anyone help to explain what is meant by this statement? get UTC time in PHP How to make clickable links and cross-references in pdfs produced by LaTeX? How can I make things link url references, my index, table of contents, and cross references clickable links in documents produced by LaTeX? I generally use pdflatex to produce my documents. I'm downloading GTA V. After 20gb / 100gb I paused my download for restarting my pc. Then download restarts from 0%.  Why this could happen? Can I restart my PC during downloads without losing progress?