Limit sum problem

Question

$a>1$ 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.

Answer 1

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 $$ \lim_{n\to+\infty}\frac{b_n}{c_n}=\lim_{n\to+\infty}\frac{b_{n+1}-b_n}{c_{n+1}-c_n}=\frac{1}{a-1}. $$