\documentclass[a4paper,12pt]{article}
\newcommand\ds{\displaystyle}
\begin{document}

\parindent=0pt

QUESTION

Determine whether the series $\ds\sum_{k=0}^\infty\frac{3^k}{k!}$
is convergent.

\bigskip

ANSWER

$\ds\sum_{k=0}^\infty\frac{3^k}{k!},\\ u_k=\frac{3^k}{k!}\\
\frac{u_{k+1}}{u_k}=\frac{3^{k+1}}{(k+1)!}\frac{k!}{3^k}=
\frac{3}{k+1}=\frac{\frac{3}{k}}{1+\frac{1}{k}}\to\frac{0}{1}=0$
as $k\to \infty$

Therefore the limit $<1$ so the series is convergent.





\end{document}
