Simple Ordinary Differential Equation
Initial value problem $t\frac{dx}{dt}=x+\sqrt{t^2+x^2}$
Evalute $ \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)} $ Evaluate $\displaystyle \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)}$.   $\bf{My\; Try::}$ Although we can solve it by converting into definite Integration.  But I want to solve it without Using Integration.  So $\displaystyle \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)} = \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{(n-1)(n-2).......(n-k+1)}{k!\cdot n^{k}\cdot (k+3)}$  Now How can i solve after that, Help required, Thanks Sum of a power series $n x^n$ I would like to know: How come that  $$\sum_{n=1}^\infty n x^n=\frac{x}{(x-1)^2}$$  Why isn't it infinity? How can I recover a deleted file on Android? I just deleted a file on my Nexus S that I really need back. (It was actually part of an app using a sqlite database that did a drop table when upgrading to a new version.)  Presumably I need to root the phone first (which I hear is easy on Nexus phones) but am I in a catch-22 here? Namely, does rooting the phone mean wiping/overwriting the partition where the data lives?  If not, what do you recommend for data recovery tools? Presumably the file was just unlinked and not actively overwritten (per usual when "deleting" a file) so technically the data must be there... Integrable function whose Fourier transform is not integrable Cauchy-Schwarz inequality proof (but not the usual one) Before you downvote/vote-to-close, I am not asking for a proof of: $$\sum^n_{i=1}a_ib_i\le\sqrt{\sum_{i=1}^na_i^2}\sqrt{\sum^n_{i=1}b_i^2 } $$ Which is what EVERY link I've found assumes is the inequality (for a proof of that see:  )    I am reading a book that claims the Cauchy-Schwarz inequality is actually: $$\vert\langle x,y\rangle\vert\le\Vert x\Vert\Vert y\Vert$$ where $\Vert x\Vert :=\sqrt{\langle x,x\rangle}$  with the additional claim: equality holds $\iff\ x,y$ are linearly dependent  I cannot find a proof of this claim (only proofs for the dot product inner product). My question is: what is the simplest way to prove this (I define simplest to mean "using as few definitions outside of the inner product itself as possible" - so that's not including Gramm-Schmitt orthonormalisation, for example.)  I've just found some possible answers in the links on the right (annoyingly) but fortunately I have a second, albeit softer, question:  What inequalities that are common for say $\mathbb{R}^n$ are actually based on the inner product?   (In future I will check this site first, I was targeting lecture notes and such in my search)  Addendum: I want to use this on not-finite dimensional vector spaces. I've found one proof that relies on finite-ness. Divergence of $\frac{ \hat {\bf r}}{r^2} \equiv \frac{{\bf r}}{r^3}$, what is the 'paradox'? One of us is wrong, aren't we? I have just learned from what I consider a reliable source, that the following sentence is correct:     One of us is wrong, aren't we?   I would never in my life have written this, but I am assured that   As I realize comments don't live forever I will quote the relevant parts:     oerkelens : So you  would really write One of us is wrong, aren't we? I guess by analogy you would not bat an eyelid at One of these balls is blue, aren't they?   Matt Эллен : yes, "Then one of us is wrong, aren't we?" is exactly how it would be written. Same for the balls.   Could someone please enlighten me how it is possible that the number in a question tag supposedly has to be in discordance with the subject of the main clause?  I admit that I am not the youngest any more, and my school days are long gone, but back in the days, I was taught that  1. a verb and its subject concord in number 2. a question tag concords with the subject of the main clause  I was given examples like:     It is warm today, isn't it?   We will be on time, won't we?   Mary is pretty, isn't she?   John isn't the brightest, is he?   Some people may notice that the subject of the main clause seems to determine every time the subject of the question tag. When the main clause subject is singular, so are verb and subject in the question tag.  I was under the impression that a) this made sense and b) this would be a general rule.  I fully realize that grammar does not have to make sense, so a) is immaterial. As for b), today I learned I was wrong :)  So when (and possibly why?) do we form question tags that are different in number from their main clause?   To exemplify, also the other example sentence that I used and was corrected on:     One of these balls is blue, isn't it? (so this is wrong)   One of these balls is blue, aren't they? (and this is correct)   As an afterthought, does this strange grammatical number mix-up only appear in question-tags, or should I always refer to singular subjects in the plural if certain conditions are met? And what are those conditions?  Which versions are correct?     One of the cars is broken, aren't they? They (the car(s?) that is(are?) broken) should be fixed.   One of the cars is broken, aren't they? It (the car that is broken) should be fixed.   One of us must be wrong, mustn't we? We (the one(s?) that is (are?) wrong) should make amends.   One of us must be wrong, mustn't we? He (the one that is wrong) should make amends.   (In the last one, they could of course be used as the singular they, but that would avoid the issue...) How can it be that the beginning universe had a high temperature and a low entropy at the same time? The Big Bang theory assumes that our universe started from a very/infinitely dense and extremely/infinitely hot state. But on the other side, it is often claimed that our universe must have been started in a state with very low or even zero entropy.   Now the third law of thermodynamic states that if the entropy of a system approaches a minimum, it's temperature approaches absolut zero.  So how can it be that the beginning universe had a high temperature and a low entropy at the same time? Wouldn't such a state be in contradiction to the third law of thermodynamics? Interpretation of simple predictions to odds ratios in logistic regression I'm somewhat new to using logistic regression, and a bit confused by a discrepancy between  my interpretations of the following values which I thought would be the same:    exponentiated beta values  predicted probability of the outcome using beta values.    Here is a simplified version of the model I am using, where undernutrition and insurance are both binary, and wealth is continuous:  Under.Nutrition ~ insurance + wealth   My (actual) model returns an exponentiated beta value of .8 for insurance, which I would interpret as:     "The probability of being undernourished for an insured individual is .8 times the probability of being undernourished for an uninsured individual."   However, when I calculate the difference in probabilities for individuals by putting in values of 0 and 1 into the insurance variable and the mean value for wealth, the difference in undernutrition is only .04. That is calculated as follows:   Probability Undernourished = exp(β0 + β1*Insurance + β2*Wealth) /                              (1+exp(β0 + β1*Insurance + β2*wealth))   I would really appreciate it if someone could explain why these values are different, and what a better interpretation (particularly for the second value) might be.     Further Clarification Edits As I understand it, the probability of being under-nourished for an uninsured person (where B1 corresponds to insurance) is:  Prob(Unins) = exp(β0 + β1*0 + β2*Wealth) /               (1+exp(β0 + β1*0+ β2*wealth))   While the Probability of being under-nourished for an insured person is:  Prob(Ins)= exp(β0 + β1*1 + β2*Wealth) /            (1+exp(β0 + β1*1+ β2*wealth))   The odds of being undernourished for an uninsured person compared to an insured person is:  exp(B1)   Is there a way to translate between these values (mathematically)? I'm still a bit confused by this equation (where I should probably be a different value on the RHS):  Prob(Ins) - Prob(Unins) != exp(B)   In layman's terms, the question is why doesn't insuring an individual change their probability of being under-nourished as much as the odds ratio indicates it does?  In my data, Prob(Ins) - Prob(Unins) = .04, where the exponentiated beta value is .8 (so why is the difference not .2?) How to read value of variable, where the name of variable is the value of another variable I think an example is in order. Consider I have these two environment variables setup (by a Continuous Delivery tool): PACKAGE_VER=1.2.3 and USE_VERSION_FROM=PACKAGE_VER.  What I want now is a script that can't predict the name of the variable pointing out 1.2.3, only USE_VERSION_FROM. And so, I'm looking for a way to get my script to know the version to use (1.2.3) without knowing anything other than that USE_VERSION_FROM points out the name of the variable that holds the version number.  My naïve non-working first attempt:  #!/bin/bash VERSION_TO_USE="${${USE_VERSION_FROM}}" Refused entry to UK, not sure what to do I'm a writer and theater/film director who has been traveling nearly all my life without incident. In February, I decided to pull up stakes in NYC and move to London. I am a freelancer and I'm working with three different organizations in NYC so I wasn't working illegally in the UK. I was, however, seeing if there might be some opportunities in London, ie. meeting with Artistic Directors of various theatre companies to see if they might be interested in producing my work.   In April, I was invited to a conference in San Diego and then a film festival in Switzerland. I didn't think it would be a problem if I went and then bummed around in France for a few days since I hadn't broken any laws and had left the UK well under my six month leave to stay. Well, when I tried to return to London, I was denied entry at Luton. The border control officer said I didn't have enough funds for a two month stay and put a black cross over a stamp in my passport. They put me in a detention room until the next flight back to Lyon, which was the following morning. I don't know anyone in Lyon so I ended up going to a friend's place in Paris.   After a few days of not knowing what to do, a friend suggested that perhaps I should try going back to London on the Eurostar since border control is in Paris. I wondered if I ought to go to the British Embassy but was deterred by something on their website that says that a visitors visa could take up to three weeks (I can't sit around here in Paris for three weeks) and anyway, the decision ultimately is with the border control, so I decided to take a chance on the Eurostar. I bought myself a return ticket in two weeks to Paris just so I had something to show.   Well, the UK border control denied me entry after giving me a third degree. They accused me of lying and seeking work in the UK. They also insisted that the amount of money I had was not enough to live on for two weeks. So I'm back at my friend's place in Paris, wondering what to do.   Is it possible for me to ever get back to the UK? I had to cancel one meeting and I'm wondering if I have to cancel going to the Sheffield DocFest. (Two of my projects were accepted to programs there.)   Should I apply for a visa from the British Embassy? Will this really take three weeks? When I started to say something about obtaining a visa, the border control officer cut me off and said, "Why are you going to the Embassy? This has nothing to do with the Embassy. We make the decisions." I've seen the word static used in different places in C code; is this like a static function/class in C# (where the implementation is shared across objects)? How should a family with young children apply for UK visas? When creating a question/answer on the Android app, it gives you a little bar above the keyboard to bold/italicize, add a link easily, add a picture etc.  When editing however, this bar isn't shown. To ease editing on mobile, it would be exceedingly helpful to have this bar show up. I find myself resorting to the full site to add a embedded link if I can't at the moment remember the exact syntax. Is it true every ideal of $\mathbb{Z}[x]$ is finitely generated? Let $I$ be any ideal of $\mathbb {Z}[x]$, and $I=\left&lt;{f_1,f_2 , ...}\right&gt;$.  Without loss of generality, $f_1$ has minimal (absolute value) coefficient of constant term (if there exist).  Then, let $g$ be any element of $I$ .  We have for some integer $m$ that $g - mf_1$ is element of $I$ (minimal coefficient and division algorithm).  I guess this process terminate finitely, but I don't know how can I do next step.  ( ex. $g-mf_1, g-mf_1-nf_2, ...$)  Please help me and thanks for in advance. I tried to choose a answer for a question on SA and someone immediately answered with the correct response. I try to click the check box and it says wait 2 min. I keep clicking until it changes from 2 min to 1 min. I wait 60 seconds on my watch and click again and it says please wait 59 seconds.  The 2 minute mark should have said 3 and the 1 minute mark was really 2.  It seems that this has , which is a lot more inconvenient than 59 seconds. Is there a public API for Stack Overflow? How to break multi digit number into separate digits in VHDL? I found the method .But I do not know how to perform this in VHDL. Let a fixed point number (12 downto -19) like 3456.478396 I need break this number entirely into separate numbers 3456.478396 --> 3,  4, 5,  6,  (dot) ,  4,  7,  8,  3,  9,  6...  into 11 integer type variables. How to perform this in VHDL?  signal example : sfixed(4 downto -4); --'example' contains 9.75 then it is storage as "01001.1100". --simply example ="10011100". --10011100 --&gt; integer 156   How to get back original 9.75 number from 156 to separate into single digits?  I USED,  library ieee; library ieee_proposed; use ieee.std_logic_1164.all; use ieee.numeric_std.all; use ieee_proposed.float_pkg.ALL; use ieee_proposed.fixed_pkg.ALL; use ieee_proposed.fixed_float_types.ALL; How do I get and modify the source code of packages installed through apt-get? How can I transfer large files from Linux to iPad over a cable or wifi? I am looking for a way to transfer large files to the iPad without using iTunes or pushing to and pulling from a remote server.  I run an Ubuntu laptop as my only OS at the moment and want to be able to transfer movies to my iPad.  I currently transfer most of my files by way of the dropbox app which works well for small files but is quite slow for larger files and eats up my bandwidth.  Ideally I'd like to be able to transfer over a cable or through WiFi.  Thanks for any tips.