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The expected value of $1/(1+X)$ where $X$ has geometric distribution Let $X$ be a random variable with a geometric distribution with parameter $p$.       Calculate $\Bbb E[1/(1+X)]$   I've looked at answers online but still do not understand how to solve this.   Specifically, I do not understand how to to get from $$\frac{p}{1-p} \sum_{x=0}^\infty (1/(1+x))((1-p)^x+1)$$  to  $-p\ln(p)  /(1-p)$. Assign material to part of mesh unbalaced data set in classification tasks Convergence of a Harmonic Continued Fraction Check if this proof about real numbers with an irrational product is correct. Can anyone confirm if my proof is correct, please?  Claim:- “If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.”  Proof:-   Assume that both $x$ and $y$ are rational. Now, let $x = \dfrac pq$ and $y = \dfrac mn$ since both of them are rational. $xy =\dfrac pq * \dfrac mn = \dfrac{pm}{qn}$ Thus, if the product $xy$ can be written as a fraction, it's not a irrational number. Therefore if one of $x$ and $y$ is not irrational, then the product is not irrational.  By the principle of proof by contraposition, If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational. What exactly do the colors in htop status bars mean? Confidence interval around binomial estimate of 0 or 1 What is the best technique to calculate a confidence interval of a binomial experiment, if your estimate is that $p=0$ (or similarly $p=1$) and sample size is relatively small, for example $n=25$? Regular expression to match a line that doesn't contain a word I know it's possible to match a word and then reverse the matches using other tools (e.g. grep -v). However, is it possible to match lines that do not contain a specific word, e.g. hede, using a regular expression?   Input:  hoho hihi haha hede   Code:  grep "&lt;Regex for 'doesn't contain hede'&gt;" input   Desired output:  hoho hihi haha Check if a services is installed using C How can I transition height: 0; to height: auto; using CSS? How to obtain and use the .bbl file in my tex document for ArXiv submission? I am submitting my first article to ArXiv and in the instructions they specify I have to include the .bbl file. Their instructions are:  "We do not run BibTeX in the auto-TeXing procedure. If you use it, include in your submission the .bbl file that BibTeX produces on your own machine; otherwise your references will not come out correctly. We do not run BibTeX because the .bib database files can be quite large, and the only thing necessary to resolve the references for a given paper is the .bbl file. The name of the .bbl file must match the name of the main .tex file for the system to process the references correctly."  My knowledge of LaTeX is basic, and I have not heard of .bbl file before. I am using TeXmaker. I have my tex file and my bib file. So what do I exactly do to obtain the .bbf file correctly and submit the article?  Thanks a lot! matlab based program where output comprises of sum of rows of input What made the Ministry admit Sirius was innocent? How do I animate time? Diamond moderator privileges vs. 10k rep privileges? As we are in election period, I don't get the difference between getting to 10k reputation point and the diamond. What additional controls do you earn with the diamond ? Possessive of a word that is already possessive As far as I know, those are the only two.  They should be pronounced Soogher and Soor, shouldn't they?  I looked them up on Dictionary.com, and their etymologies reveal no trace of an SH, except where the listing for sugar had:     Middle English sugre, sucre (noun) &lt; Middle French sucre  &lt; Medieval Latin succārum  &lt; Italian zucchero  &lt; Arabic sukkar;  obscurely akin to Persian shakar, Greek sákcharon   I see an obscure kinship to shakar, but the word morphed so many times since then that the SH disappeared pretty much completely.  And sure is even worse, with no sign of SH:     Middle English sur ( e ) &lt; Middle French sur, Old French seur  &lt; Latin sēcūrus   Why are these the only two like this? Slovak (and Czech) babel gives problems with cmidrule and cline I tried to use the Slovak language with the booktabs package.    Let's see a couple of cases.   slovak is included in the options to babel     \usepackage[english,slovak]{babel}   or    \usepackage[slovak,english]{babel}   or    \usepackage[slovak]{babel}`  "slovak" is NOT included in the options to babel  \usepackage[english]{babel}    Similarly two cases for \cmidrule  A. \cmidrule{2-3} is included in the table    B. \cmidrule{2-3} is NOT included in the table    If both (slovak+\cmidrule) are included (case 1A), then following error occurrs  Paragraph ended before \@@@cmidrule was complete   (and a lot of other errors)  In the other cases (1B, 2A and 2B) (NO slovak or NO \cmidrule), then pdflatex is succesful  Does anybody know why 1A is wrong?    And how to use slovak and \cmidrule together (case 1A)?  \documentclass[10pt]{book} \usepackage[english,slovak]{babel} \usepackage{booktabs}  \begin{document}  \begin{tabular}{ccc} A &amp; B &amp; C \\ %\cmidrule{2-3} A &amp; B &amp; C \\ \end{tabular}  \end{document}   I have Win7, TeXstudio 2.5.2 Structuring game narrative. Handling NPC, Quest, General Dialogue. Is Json/XML the right way to go? Or any other solutions? Hello guys I'm try to make some games that have more to it than just simple high score scrolling shooters etc.  I'm trying to design my own systm of handling narrative. I want to be able to place NPC's in the game world, with scripts attached to them. Some of the NPC's would just have n random greetings. Others will be mission givers and will offer you a quest of sorts. (The quest could be go speak to Bob in the pub, or it might be bring me the Holy Grail and Crown Jewels, or something in between).  I've wracked my brain for a long while over this but sadly keep slipping back to the only way I know how, which is to manually code all the strings that people can say and have a horrible web of if statements and other such insanity to give me a rudimental dialogue system. Its not good enough even with just a few characters, but if I had dozens/hundreds of characters it would be even worse.  I've read lots about it already but I'm still very confused when I start trying on my own project. I've read that Json/XML might be the way to go and that I can add custom markups in the Json text to call methods in my code, but I dont even know if this is right and if so how I can do it.  The back of my brain is screaming at me that there is a simpler way to do it, but I've sat here for a very long time without managing to think of ideas.  Here is a diagram I drew which pretty much shows the flow of how I want my NPC to behave. If anyone can see this and then tell me how they might do something like this that would be fantastic!! (note: if Json is the correct advise, please give me a exact example of how exactly to format the Json in order to check the bools I mention in my flowchart - thanks!):    note: The circles denote an output from the data as Text for the dialogue system. The bools will need to be checked against by the data file to see which output to give (hope that makes sense and is possible) Books for Hyperbolic Geometry. I want to read hyperbolic geometry. Can any one suggest some good books on the topic.