Limit sum problem $a>1$
Can anyone help me with this limit problem.
I think I sholud use Cesaro - Stolz theorem but, I don't know how.
 
Wihout using L'Hospital rule and integration.

Let
$$
b_n=a+\frac{a^2}{2}+\cdots+\frac{a^n}{n}
$$
and
$$
c_n=\frac{a^{n+1}}{n}.
$$
$c_n$ will be monotonically increasing to $+\infty$ (at least from some $n$ on) since $a>1$. Now I suggest you to study the limit of
$$
\frac{b_{n+1}-b_n}{c_{n+1}-c_n}
$$
and then apply the theorem of Stolz–Cesàro. I will stop here, and hope that you can conclude. If not, I will happily give the full calculations of the limit in the displayed equation above.
Edit with more steps
We get
$$
b_{n+1}-b_n=\frac{a^{n+1}}{n+1}
$$
and
$$
c_{n+1}-c_n=\frac{a^{n+2}}{n+1}-\frac{a^{n+1}}{n}.
$$
Thus,
$$
\frac{b_{n+1}-b_n}{c_{n+1}-c_n}=\frac{\frac{1}{n+1}}{\frac{a}{n+1}-\frac{1}{n}}=\frac{1}{a-1-\frac{1}{n}}\to\frac{1}{a-1}
$$
as $n\to+\infty$. An application of Stolz–Cesàro gives
$$
\lim_{n\to+\infty}\frac{b_n}{c_n}=\lim_{n\to+\infty}\frac{b_{n+1}-b_n}{c_{n+1}-c_n}=\frac{1}{a-1}.
$$
