\displaystyle \lim_n {\frac{n}{a^{n+1}} ( a+\frac{a^2}{2} + \cdots+ \frac{a^n}{n})} =
\lim_n {\frac{ a+\frac{a^2}{2} + \cdots +\frac{a^n}{n} }{\frac{a^{n+1}}{n}}}