Limit with factorial and summation
Evalute $ \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)} $
Voldemort was said to be very interested in the four founders of Hogwarts. He created a Horcrux from the Ravenclaw diadem, the Hufflepuff cup, and the Slytherin locket. Did he try to find a Gryffindor relic to complete the set, or did he not bother?   I can't see him giving up on such a task, but maybe there was insufficient time.  Gryffindor's sword was hiding 'inside' the Sorting Hat, but a famous family usually has some other heirlooms too.  If there are no canon answers I'd be interested in your own speculation! Solidify conundrum? I stumbled across a solidify issue, I cannot wrap my head around. I have a simple mesh, consisting of three faces.  When adding a solidify modifier, I expect to get an object with three walls, where every angle is square. However, whatever I try, tweaking the various solidify parameters, what I get is something like the following, with a skewed rim.  Did I hit a bug, or is it a limitation of the solidify modifier? Is there a workaround? Even when /tmp has no file called something, searching for it with find will return 0:    $ find /tmp -name something    $ echo $?   0   How can I get a non-zero exit status when find does not find anything? When I type uname -a, it gives the following output.  Linux mars 2.6.9-67.0.15.ELsmp #1 SMP Tue Apr 22 13:50:33 EDT 2008 i686 i686 i386 GNU/Linux   How can I know from this that the given OS is 32 or 64 bit?  This is useful when writing configure scripts, for example: what architecture am I building for? I am working on a game and would like to include a few jump scares. So here is a rather simple question — What makes a good jump scare and when and how should they happen? Changing threshold in logistic regression Suppose that you have a logistic model and the predicted probabilities of a $1$ all are in $[0.2,0.3]$. So the predictions would be "0" based on a threshold of $0.5$. What would be a good threshold to choose? Is there a GLM bible? Is there consensus in the field of statistics that one book is the absolute best source and completely covering every aspect of GLM - detailing everything from estimation to inference? I have a php file which I will be using as exclusively as an include. Therefore I would like to throw an error instead of executing it when it's accessed directly by typing in the URL instead of being included.  Basically I need to do a check as follows in the php file:  if ( $REQUEST_URL == $URL_OF_CURRENT_PAGE ) die ("Direct access not premitted");   Is there an easy way to do this? Why are irrational numbers uncountable and rationals countable? Question 1: Why are irrational numbers uncountable and rationals countable?  I really struggle to understand this. I initially thought it had something to with the fact that between any two numbers there will always be a irrational number. But now it has come to my attention that the same holds for rational numbers. So I can't really see the difference between them. What is it that makes the difference? Why can we line up all rational numbers one-to-one with the natural numbers, but not the irrationals with the naturals?  Sorry for a perhaps basic question, but I really, really struggle with it. Thanks. How can I account for a nonlinear variable in a logistic regression? How can I account for a nonlinear independent variable in a logistic regression?  For example, consider this data set:  b1  b2  b3  b4  b5 1   1   0   0   20 0   1   0   0   20 1   0   0   0   13 1   1   1   0   8 1   1   0   1   5 0   1   0   1   4 1   1   0   1   5 0   0   0   0   8 0   0   1   0   13 1   0   0   1   20 1   0   0   0   29 1   1   0   1   40 1   0   0   0   53 0   1   1   0   68   Suppose I gathered some data like the above (suppose I had 10,000 rows) and I want to use them to predict future sets of similar data with an equation generated by a logistic regression. I get the above data. My independent variables are b1 through b5. By looking at the data, I can clearly see that independent variables b1 through b4 are binary (maybe they are male/female, rent/own, etc).   However I see that b5 is clearly not binary. Upon graphing it and looking at the numbers I determine that the b5 variable has a bathtub shaped U curve with an equation of b5=(x-4)^2+4.   I understand that using the b5 variable as it is would screw up my logistic regression - since any interval independent variables must have a linear relationship with the dependent variable.   Yet, I must account for the variable b5 in my data set while doing a logistic regression for it to be accurate and predict well (right?).  Given the above data, how would I go about making a predictive model with a logistic regression? Any references online or in books would greatly be appreciated as I am having trouble finding a good explanation of the above.  Cheers =)  EDIT:  Sorry if my question is unclear.  Consider this: Suppose we had the following data on many people:  Y: Whether they had a stroke before the age of 50 or not 1. Whether they own a pet or not. 2. Whether they graduated college or not. 3. Whether they are male or female. 4. The number of hours they exercise every week.  We want to train a logistic regression model based on historical data to predict if someone in the future with the same data will be more apt or not to have a stroke before the age of 50.  The problem I am having is with the last variable. A continuous ratio variable like "number of hours exercised each week."  Exercising more each week may have diminishing returns on health and thus having a stroke. For example decreasing an increasing rate. Or maybe (totally made up) exercising a little bit is really good for you, exercising a moderate amount is really bad for you, and exercising a lot is really good for you again. We might have a U shaped curve like (x-4)^2+4 for example when plotting health to exercise.   I would image that having a curved, non-linear independent variable in a logistic regression like the one I described above would cause problems. Am I right? And if so, what kinds of things can you do to still include the variable in the logistic regression. Perhaps a transformation? Is that all? How to apply for a Schengen visa when the travel dates are uncertain My husband has a Chinese passport and I am a U.S. citizen; I don't need a visa to travel to Schengen countries. I am helping him to apply for a visa.  Our plans are to go to Paris, Italy and Spain, but we haven't nailed down definite dates; the approximate dates are August 18 to September 11. We made an appointment with Spain consulate to apply for the visa. We need to show air flight reservations, I could book it online without any problems.    Since we don't have definite dates, what dates should he put down on the application?  I know that, once they approve you, they give you 90 days, so does it matter what dates I put down on the visa application?   Our appointment at the Spanish Consulate is in a few days. I'm very familiar with how symbols display differently in inline mode (with $ ... $) vs. how display math shows them (with \[ ... \] or $$ ... $$).  Two examples would be with limits  $\lim_{n\rightarrow \infty}f(x)$   vs.  \[ \lim_{n\rightarrow \infty}f(x) \]   and with sums  $\sum_{n=1}^{x} n^2$   vs.  \[ \sum_{n=1}^{x} n^2 \]   My Question is:     How do you display these things in inline mode (with all the nice inline formatting) as if they were in display mode ?   The inverse -- how to display things in display mode as if they were in inline mode -- is also of interest. What are the advantages of 10-bit monitors? To support 10-bit color the following are needed:   A monitor supporting it. A GPU supporting it (only AMD FirePro and NVIDIA Quadro support this?). Compatible software. Unless I am mistaken there are very few programs out there supporting 10-bit color. Photoshop is a notable example.   The questions are about how 10-bit monitors perform in comparison with 8-bit monitors:   In which situations would a 10-bit monitor give a noticeable advantage over an 8-bit monitor (say, for professional photography)? Have 10-bit monitors been compared against 8-bit monitors based on subjective or objective tests? What were the results? Human eyes can see only 10m colors, so would using a monitor with 1b colors make a difference? What does the explicit keyword mean? What does the explicit keyword mean in C++? What ensures big holes in my bread? I've got a NI Traktor Audio 2 sound card I'd like to get working with Mixxx, but I don't seem to have any success so far. The card seems to be supported in Ubuntu out of the box, and it seems to work well with ALSA, but I cannot select it in Mixxx and I'm not sure whether is something I can do to make it see it (e.g. through).  So for testing purposes, on the command line I can execute aplay to check each channel:  aplay -D plughw:2,0,0 test.wav   and  aplay -D plughw:2,0,1 test.wav   That works well: I can hear the test sound on each channel (A and B)  The relevant output for this card using aplay -D is:  hw:CARD=TraktorAudio2,DEV=0      Traktor Audio 2, Traktor Audio 2      Direct hardware device without any conversions plughw:CARD=TraktorAudio2,DEV=0      Traktor Audio 2, Traktor Audio 2      Hardware device with all software conversions   From those 2 items on the output, plughw is the one that works, whereas hw doesn't.  The problem is that Mixxx only lets me select the hw card, which does not seem to work. I'd like to be able to select the plughw card, which is the one that's working.  I've tried suspending Pulse Audio as suggested by , but that did not seem to work. I also played around with an .asoundrc file similar to the one  (which was for an older model of this sound card), but also without success.  Any ideas on how to make Mixxx show the right card (and channel, while we are at it)?  Thanks! How to recover a deleted NTFS partition with data I am on Ubuntu 12.04 64-bits  I accidentally deleted a NTFS 255 GB partition using Gparted, and yes it contains LOTS of important data.  Upon realizing the deletion I haven't yet created any file-system on it. So it still shows Unallocated.    I tried Testdisk , but it shows entire partitions created from the beginning.  This sdb is my secondary drive with Xubuntu and Kubuntu on it. And that 255 GB NTFS  partition was cross platform storage drive.  I am currently in sda with Ubuntu and Win7 in dual boot, which enables me to use it as Safe recovery option rather then Live USB drive.   First ,  should i allocate it to ? file-System for better recovery .  Second , i really need to recover data from this drive so please give me possible  suggestions and proven methods of data recovery.  Third ,does Testdisk requires deleted partition to be allocated for its recovery .  I am currently trying the answer seen here  with Testdisk scanning for partition second time, hoping to recover it.   It looks a duplicate but i can't confirm if the OP got it right. Since my first doubt is to allocate it or not.  If any clarification required please do comment ,i will add it.  Edit: TestDisk deeper search results.      The following partitions can't be recovered:          Partition               Start        End    Size in sectors     &gt;  HPFS - NTFS          22241 107 11 55592   3 27  535777280        FAT32 LBA            274460 158 19 460576 228 16 2989957948  NTFS, 274 GB / 255 GiB &amp; for FAT32 LBA 1530 GB / 1425 GiB Proof of $1+x\leq e^x$ for all x? gnome-desktop-item-edit: command not found on Ubuntu 19.10 and later even with the 'gnome-panel' package installed If $F_n$ is the $n$-th fibonacci number, then prove that, $$F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$$  I tried the idea of using Pascal's triangle, but it seems to need some adjustment, so I am stuck, please help.    Any combinatorial argument would be of greater help than rigorous proof.  Thank you.