%% %% This is file `exfsamp.tex', %% generated with the docstrip utility. %% %% The original source files were: %% %% exframe.dtx (with options: `samplesingle') %% %% Copyright (C) 2011-2020 Niklas Beisert %% %% This work may be distributed and/or modified under the %% conditions of the LaTeX Project Public License, either version 1.3 %% of this license or (at your option) any later version. %% The latest version of this license is in %% http://www.latex-project.org/lppl.txt %% and version 1.3 or later is part of all distributions of LaTeX %% version 2005/12/01 or later. %% \NeedsTeXFormat{LaTeX2e}[1996/12/01] \ProvidesFile{exfsamp.tex}[2020/02/24 v3.4 standalone sample for exframe] \documentclass[12pt]{article} \usepackage{geometry} \geometry{layout=a4paper} \geometry{paper=a4paper} \geometry{margin=2.5cm} \parindent0pt \parskip0.5ex \usepackage{amsmath} \usepackage{hyperref} \PassOptionsToPackage{loadlang=en|de}{metastr} \PassOptionsToPackage{course=true}{metastr} %%\usepackage{metastr} %%\metasetlang{de} \usepackage[extstyle]{exframe} %%\exercisesetup{solutions=true} \exercisesetup{solutions=false} \exercisesetup{autolabelproblem=true} \exercisestyle{plainheader} \exerciseconfig{composeheaderbelowright}{\getsheetdata{points}}% \exerciseconfig{countersheet}{\Roman{sheet}} \exerciseconfig{countersubproblem}{\roman{subproblem})} \exerciseconfig{countersubproblemmax}{vii)} \exerciseconfig{insertsubprobleminfo}{% \switchpoints{}{\addprobleminfo*{% \hspace{-\getexerciseconfig{skipsubprobleminfo}}*}}% {}{}{\getsubproblempoints{}}} \ifdefined\metaset \metasetterm[en]{sheet}{Exercise Sheet} \metasetterm[en]{sheets}{Exercise Sheets} \metasetterm[de]{sheet}{\"Ubungsblatt} \metasetterm[de]{sheets}{\"Ubungsbl\"atter} \else \exerciseconfig{termsheet}{\"Ubungsblatt} \exerciseconfig{termsheets}{\"Ubungsbl\"atter} \fi \exercisestyle{problempointsat=margin} \reversemarginpar \exerciseconfig{composepointsmargin}[1]{#1p.} \exerciseconfig{composepointspairmargin}[2]{% \ifdim#2pt=0pt#1p.% \else\ifdim#1pt=0pt+#2p.% \else#1+#2p.% \fi\fi} \exerciseconfig{styletitle}{\sffamily\bfseries} \exerciseconfig{skipproblembelow}{1.5cm} \exercisestyle{fracpoints} \exercisestyle{solutionbelow=problem} \exercisestyle{solutionsep} \exercisesetup{pdfdata=sheet} \ifdefined\metaset \metaset[sep]{subtitle}{, } \metaset{subtitle}{\ifsolutions\metatranslate[#1]{solutions} \fi% \metaif[use]{sheettitle} {\metapick[#1]{sheettitle}} {\metapick[#1]{material}}} \metaset{author}{\exerciseifempty{\getsheetdata{author}}% {\metapick[#1]{instructor}}{\metapick[#1]{sheetauthor}}, \metapick[#1]{institution}} \else \exerciseconfig{composemetasheet}[2]{\getexercisedata{course}, \ifsolutions\getexerciseconfig{termsolutions} \fi% \exerciseifempty{#2}{\getexerciseconfig{termsheet} #1}{#2}} \exercisedata{title=% {\getexercisedata{course}, \ifsolutions\getexercisedata{solutions} \fi% \getexercisedata{material}}} \exercisedata{author=% {\getexercisedata{instructor}, \getexercisedata{institution}}} \fi \ifdefined\metaset \metaset{institution}{Katharinen-Volksschule} \metaset[de]{course}{Mathematik} \metaset[en]{course}{Mathematics} \metaset{instructor}{J.\ G.\ B\"uttner} \metaset{period}{ca.\ 1786} \metaset[de]{material}{\"Ubungsaufgaben} \metaset[en]{material}{Exercise Problems} \else \exercisedata{institution={Katharinen-Volksschule}} \exercisedata{course={Mathematik}} \exercisedata{instructor={J.\ G.\ B\"uttner}} \exercisedata{period={ca.\ 1786}} \exercisedata{material={\"Ubungsaufgaben}} \fi \begin{document} \begin{sheet}[number=5,label={sheet5}] \begin{problem}[title={Sums},points=99+4] \exerciseloopstr{\getsubproblemlist{}}{c}% \hfill\begin{tabular}{c|\exerciseloopret|c} \exerciseloop{\getsubproblemlist{}} {&\ref{\getexerciseconfig{labelsubproblem}{#1}}} &\ref{prob:\problemtag}\\\hline \getexerciseconfig{termpoints} \exerciseloop{\getsubproblemlist{}}{&\extractpoints{\getsubproblempoints{#1}}} &\extractpoints{\getproblempoints{}} \\ extra \exerciseloop{\getsubproblemlist{}}{&\extractpoints*{\getsubproblempoints{#1}}} &\extractpoints*{\getproblempoints{}} \end{tabular} This problem deals with sums and series. \begin{subproblem}[points=2,difficulty=simple,label={\problemtag-simplesum}] Compute the sum \showpoints \begin{equation} 1+2+3. \end{equation} \begin{solution} The result is \begin{equation} 1+2+3=6. \end{equation} \end{solution} \end{subproblem} \begin{subproblem}[points=97+0.5,difficulty=lengthy] Compute the sum \begin{equation} 1+2+3+\ldots+98+99+100. \end{equation} Keep calm and calculate! %%That ought to keep him occupied for a while \end{subproblem} \begin{solution}[author={C.\ F.\ Gau\ss}] We use the result $1+2+3=6$ from part \ref{\problemtag-simplesum} to jumpstart the calculation. The remaining sums yield \awardpoints*[1 for each remaining sum]{97} \begin{equation} 6+4+5+\ldots+99+100=5050. \end{equation} Alternatively the summands can be grouped into pairs as follows: \begin{align} 1+100&=101,\\ 2+99&=101,\\ 3+98&=101,\\ \ldots &\nonumber\\ 50+51&=101. \end{align} These amount to 50 times the same number 101. Therefore the sum equals \begin{equation} 1+2+\ldots+99+100=50\cdot 101=5050. \end{equation} \textit{Ligget se!} \awardpoints{97+0.5} \end{solution} You may give the final part a try: \begin{subproblem}[optional={optional}, difficulty={requires inspiration},points={+3.5}] Compute the series \showpoints \begin{equation} 1+2+3+\ldots \end{equation} \begin{solution} The series is divergent, so the result is $\infty$ \awardpoints{+1}. \par However, after subtracting the divergent part, the result clearly is \begin{equation} \zeta(-1)=-\frac{1}{12}\,, \end{equation} where the zeta-function $\zeta(s)$ is defined by \begin{equation} \zeta(s):=\sum_{k=1}^\infty \frac{1}{k^s}\,. \end{equation} This definition holds only for $s>1$ where the sum is convergent, but one can continue the complex analytic function to $s<0$ \awardpoints{+1.5}. \par Another way of understanding the result is to use the indefinite summation formula for arbitrary exponent $s$ in the summand (which also follows from the Euler--MacLaurin formula) \begin{equation} \sum_n n^s = \frac{n^{s+1}}{s+1} -\sum_{j=0}^s \frac{\zeta(j-s)\,s!}{(s-j)!\,j!}\,n^j = \ldots - \zeta(-s)\,n^0. \end{equation} Curiously, the constant term with $j=0$ is just the desired result but with the wrong sign (in fact, the constant term of an indefinite sum is ambiguous; for the claim we merely set $j=0$ in the expression which holds for others values of $j$) \awardpoints{+0.5}. In order to understand the sign, we propose that the above formula describes the regularised result for the sum with limits $+\infty$ and $n$ \begin{equation} \sum_{k=+\infty}^n k^s \simeq \frac{n^{s+1}}{s+1} -\sum_{j=0}^s \frac{\zeta(j-s)\,s!}{(s-j)!\,j!}\,n^j. \end{equation} Then we flip the summation limits of the desired sum to bring it into the above form \awardpoints{+0.5} \begin{equation} \sum_{k=1}^\infty k^s = -\sum_{k=\infty}^0 k^s \simeq \zeta(-s). \end{equation} \end{solution} \end{subproblem} \end{problem} \begin{problem}[points=1, difficulty=insane] Show that the equation \begin{equation} a^3+b^3=c^3 \end{equation} has no positive integer solutions. \end{problem} \begin{solution} \normalmarginpar This is beyond the scope of this example. \marginpar{\footnotesize\raggedright does not fit here.\par} \end{solution} \ifsolutions\else \textbf{Grading:}\par \exerciseloopstr{\getproblemlist{}}{|c} \begin{tabular}{|c|\exerciseloopret||c|}\hline \getexerciseconfig{termsheet} \ref{sheet5} \exerciseloop{\getproblemlist{*}} {&\ref{\getexerciseconfig{labelproblem}{#1}}} &total \\\hline value \exerciseloop{\getproblemlist{*}} {&\extractpoints{\getproblempoints{#1}}}% &\extractpoints{\getsheetpoints{}} \\\hline \exerciseloop{\getproblemlist{*}}{&} &\\\hline \end{tabular}\qquad \exerciseloop{\getproblemlist{*}}{ \exerciseloopstr{\getsubproblemlist{#1}}{|c} \ifnum\value{exerciseloop}>0\relax \begin{tabular}{|c|\exerciseloopret||c|}\hline \getexerciseconfig{termproblem} \ref{\getexerciseconfig{labelproblem}{#1}} \exerciseloop{\getsubproblemlist{#1}} {&\ref{\getexerciseconfig{labelsubproblem}{##1}}} &total \\\hline value \exerciseloop{\getsubproblemlist{#1}} {&\extractpoints{\getsubproblempoints{##1}}}% &\extractpoints{\getproblempoints{#1}} \\\hline \exerciseloop{\getsubproblemlist{#1}}{&} &\\\hline \end{tabular}\quad \fi } \fi \end{sheet} \end{document} \endinput %% %% End of file `exfsamp.tex'.