calculate the given limites ?1

Question

calculate

$\lim_{n\to\infty}\frac{n}{a^{n+1}}$( $a$+$\frac {a^2}{2} +.....+\frac{a^n}{n}$), where $a >1$.

here i take $x_n$ = $a$ + $\frac {a^2}{2} +.....+\frac{a^n}{n}$ and $y_n$ =$\frac {a^{n+1}}{n}$

now $\lim_{n\to\infty}$ =$\frac {x_n }{y_n}$ =$\frac{n}{a^{n+1}}a$ + $\frac{n}{a^{n+1}}\frac {a^2}{2}$ +...........+$\frac{n}{a^{n+1}}$$\frac{a^n}{n}$ = $\infty$

Is its corrects ????

Pliz help me

thanks in advance

Answer 1

HINT

Let consider

$$\frac{a_n}{b_n}=\frac{\sum_{k=1}^n \frac{a^k}{k}} {\frac{a^{n+1}}{n}}$$

and apply Stolz-Cesaro

$$\lim_{n \to \infty} \frac{a_n}{b_n}=\lim_{n \to \infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n}$$