\documentclass{article} \usepackage[dvipsnames]{xcolor} \colorlet{LightRubineRed}{RubineRed!70} \colorlet{Mycolor1}{green!10!orange} \definecolor{Mycolor2}{HTML}{00F9DE} \begin{document} \pagecolor{black} \color{white}% Using Mathematical Induction, prove that : $e^{x}>1+x+{x^2\over 2!}+{x^3\over 3!}+...+{x^n\over n!}$ if $x>0$ and n is any positive integer.
\noindent {\color{LightRubineRed} \rule{\linewidth}{1mm}} \noindent {\color{RubineRed} \rule{\linewidth}{1mm}} \begin{itemize} \item \Large \itshape \textcolor{Mycolor1}{My attempt} \item \textcolor{Mycolor2}{Let} $f(x)=e^x-1-x-{x^2\over2!}-{x^3\over 3!}-...-{x^n\over n!}$
\item \textcolor{Mycolor2}{Then need to find} ${d\over dx}f(x)$ \item \textcolor{Mycolor2}{and apply the inequality}$f'(x) > 0$
$f'(x)=e^x-1-x-{x^2\over2!}-{x^3\over 3!}-...-{x^{n-1}\over (n-1)!}$
$f'(x)=f(x)+{x^n\over n!}$\item \textcolor{Mycolor2}{I don't know how to proceed further.Any suggestions, hint would be appreciated.} \end{itemize} \end{document}