How would you evaluate the following limit as n goes to infinity?  $$\lim \frac {1}{(1+\frac {1}{n})^n}$$  I would of thought that this would evaluated to be,  $$\lim \frac {1}{(1)^n} = 0 $$  However the correct answer is $$\frac{1}{e}$$
I was wondering if it is possible to get a link to a rigorous proof that  $$\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$$
How do I test a nonlinear association? For plot 1, I can test the association between x and y by doing a simple correlation.   For plot 2, where the relationship is nonlinear yet there is a clear relation between x and y, how can I test the association and label its nature? 1.  Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an explanation that I need some help with: that the de Broglie wavelength and the wavelength of an elastic wave do not show similar properties under a Galilean transformation. He basically says that both are equivalent under a gauge transform and also, separately by Lorentz transforms. This, accompanied with the observation that $\psi$ is not observable, so there is no "reason for it being real". Can someone give me an intuitive prelude by what is a gauge transform and why does it give the same result as a Lorentz tranformation in a non-relativistic setting? And eventually how in this "grand scheme" the complex nature of the wave function becomes evident.. in a way that a dummy like me can understand.  2.   A wavefunction can be thought of as a scalar field (has a scalar value in every point ($r,t$) given by $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{C}$ and also as a ray in Hilbert space (a vector). How are these two perspectives the same (this is possibly something elementary that I am missing out, or getting confused by definitions and terminology, if that is the case I am desperate for help ;)   3.   One way I have thought about the above question is that the wave function can be equivalently written in $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{R}^2 $ i.e, Since a wave function is complex, the Schroedinger equation could in principle be written equivalently as coupled differential equations in two real functions which staisfy the Cauchy-Riemann conditions. ie, if $$\psi(x,t) = u(x,t) + i v(x,t)$$ and $u_x=v_t$ ; $u_t = -v_x$ and we get $$\hbar \partial_t u  = -\frac{\hbar^2}{2m} \partial_x^2v + V v$$ $$\hbar \partial_t v  = \frac{\hbar^2}{2m} \partial_x^2u - V u$$ (..in 1-D) If this is correct what are the interpretations of the $u,v$.. and why isn't it useful. (I am assuming that physical problems always have an analytic $\psi(r,t)$). Do the last digits of powers of a number $n$ follow the same cycle as the last digits of the number $n$'s last digit's powers? I'm creating a color object using the following code.  curView.backgroundColor = [[UIColor alloc] initWithHue:229 saturation:40 brightness:75 alpha:1];   How can I retrieve RGB values from the created color object? How to get the path of a running JAR file? My code runs inside a JAR file, say foo.jar, and I need to know, in the code, in which folder the running foo.jar is.  So, if foo.jar is in C:\FOO\, I want to get that path no matter what my current working directory is. Can Blender render pngs with the background transparent? How can I get sql string, generated by Linq2SQL ORM? Can I include another file in fstab Can I include another file in my fstab file?  I.e. can I have fstab execute lines from another file at an arbitrary point, and then return to the main file when completed? Case field mapping with custom object fields? Why doesn't 'find' show this file? Using find with grep, one can locate files that match a pattern:  # find | grep error ./solr-modifiedSolr4/SolrPhpClient/phpdocs/errors.html ./error_log ./includes/classes/error_log   However, using find alone the first file is not found:  # find . -name error* ./error_log ./includes/classes/error_log   Why doesn't find locate the errors.html file when not used with grep? How is find used to show this file as well? Can we get our personal voting data displayed on the Questions lists? There's been several times where I have been surfing the site, and noticed a question that I am not sure I've checked. Even if I've checked the question before, I can't recall off-hand whether or not I've voted on it or its answer.  Is there a way we can have our personal voting history reflected to some degree or another, in the Questions listing? For example, could there be an indicator as to whether or not we have already voted (up or down) on a question, and perhaps maybe an additional indicator as to whether we've up-voted any of its answers?  It would be very convenient to have some of this data available up-front, without actually having to view the question again. Is there an idiom for people who boast too much? I am looking for idioms or informal/slang/colloquial expression for some people that make you think that they are able of building a skyscraper, constructing a spaceship, playing the piano better than Mozart or something very fancy that requires remarkable skills, but when it comes to practice they prove to be completely inadequate. How do we know the brain flips images projected on the retina back around? I am replacing a door with a pre-hung door including the jamb. The old door was 29 11/16" wide. The new pre-hung door is 28".  Is this too big of a gap?  Should I make it plumb on one side and make up the difference on the other side? Other threads suggest making up space on the handle side, but this is more space than than what other people were asking about. The trim is 2 1/2" so the gap would be covered. Instead of relying on my host to send an email, I was thinking of sending the email messages using my Gmail account. The emails are personalized emails to the bands I play on my show.   Is it possible to do it? I was surprised to discover my dictionary had this entry for dilemma:     a situation in which a difficult choice has to be made between two or more alternatives, esp. equally undesirable ones   The notion of dilemma meaning two or more flies against what I was taught about the word. The very idea of a  is specifically based on the number two.  Has my dictionary merely updated its definition to encapsulate the many people who use dilemma for more than two equal choices? Or was someone in my youth being unnecessarily pedantic? Exercise on local and global symmetry for a given Lagrangian density I have a model which you can see here (camera perspective).    In the second image, you can see how the render turned out.    I'm at my wit's end trying to figure our why it's not showing up in the render. If anyone needs more information, just ask and I'll get it to you. I've already checked the normals and they're not the cause. the structure "like for somebody to do something" I've started putting into practice the answer to . Yet I would like to change this so that there are brackets around the citations:   in the superscript in text,     in the superscript in footnote,  in the bibliography.   I've managed to get the first one and the last one working. The second one however, I don't know how to do this.   Note that I mainly write in english and in dutch, so I don't think there is much babel interference.  Also, would there be a way to order the footnotes so that first all ''normal'' \footnote{} are printed and below that all \sfcite{} citational footnotes (per page that is)?  Here's what I've managed to concoct:  \documentclass{article} \usepackage[english]{babel} \usepackage[pdfborder={0 0 0}]{hyperref} \usepackage[citestyle=numeric-comp,sorting=none,hyperref]{biblatex} \usepackage{filecontents}  % No brackets around the number of each bibliography entry \DeclareFieldFormat{labelnumberwidth}{\textbf{[#1]}}  %--------------------------------------------------------------- % Essentially verbatim from Joseph Wright % http://www.texdev.net/2010/03/08/biblatex-numbered-citations-as-footnotes/ \makeatletter \ExecuteBibliographyOptions{citetracker,sorting=none}  \DeclareCiteCommand{\notefullcite}[\mkbibbrackets]     {\usebibmacro{cite:init}%         \usebibmacro{prenote}}     {\usebibmacro{citeindex}%         \usebibmacro{notefullcite}%         \usebibmacro{cite:comp}}     {}     {\usebibmacro{cite:dump}%         \usebibmacro{postnote}}  \newbibmacro*{notefullcite}{%     \ifciteseen         {}         {\footnotetext[\thefield{labelnumber}]{%             \usedriver{}{\thefield{entrytype}}\addperiod}}}  \DeclareCiteCommand{\sfcite}[\cbx@superscript]%     {\usebibmacro{cite:init}%         \let\multicitedelim=\supercitedelim         \iffieldundef{prenote}         {}         {\BibliographyWarning{Ignoring prenote argument}}%     \iffieldundef{postnote}         {}         {\BibliographyWarning{Ignoring postnote argument}}}     {\usebibmacro{citeindex}%         \usebibmacro{sfcite}%         \usebibmacro{cite:comp}}     {}     {\usebibmacro{cite:dump}}  \newbibmacro*{sfcite}{%     \ifciteseen         {}         {\xappto\cbx@citehook{%             \noexpand\footnotetext[\thefield{labelnumber}]{%             \fullcite{\thefield{entrykey}}\addperiod}}}}  \newrobustcmd{\cbx@superscript}[1]{%     \mkbibsuperscript{[#1]}%     \cbx@citehook     \global\let\cbx@citehook=\empty} \let\cbx@citehook=\empty \makeatother %---------------------------------------------------------------  \begin{filecontents}{\jobname.bib} @article{ref1,     hyphenation = {american},     title = {Natural Products as Sources of New Drugs over the Last 25 Years},     author = {Newman, D.J. and Cragg, G.M.},     journal = {J. of Nat. Prod.},     volume = {70},     number = {3},     pages = {461--477},     year = {2007}} @article{ref2,     hyphenation = {american},     title = {Natural products as leads to potential drugs: an old process or the new hope for drug discovery?},     author = {Newman, D.J.},     journal = {J. of Med. Chem.},     volume = {51},     number = {9},     pages = {2589--2599},     year = {2008}} @online{ref3,     author = {Quintin, C.},     title = {Les eponges},     url = {http://christophe.quintin.pagesperso-orange.fr/Liste_Doc.html},     urldate = {2009-05-03} } \end{filecontents}  \addbibresource{\jobname.bib}  \begin{document} First citation.\sfcite{ref1,ref2} Some text\footnote{A footnote}.   \pagebreak Second citation\sfcite{ref1}, second citation\sfcite{ref2}, first citation.\sfcite{ref3} \printbibliography \end{document}