check if $\sum_{k=1}^{\infty}{ \left( \frac{7k-2}{8k-3 \sqrt{k}}\right)}^k$ converges

Question

How to check if $\sum_{k=1}^{\infty}{ \left( \frac{7k-2}{8k-3 \sqrt{k}}\right)}^k$ converges?.

$\begin{align} \sum_{k=1}^{\infty}{ \left( \frac{7k-2}{8k-3 \sqrt{k}}\right)}^k &= \lim_{n \to \infty} \sum_{k=1}^{n}{ \left( \frac{7k-2}{8k-3 \sqrt{k}}\right)}^k \\ &= \lim_{n \to \infty} \sum_{k=1}^{n}{ \left( \frac{7\frac{k}{k}-2\frac{1}{k}}{8\frac{k}{k}-3 \frac{\sqrt{k}}{k}}\right)}^k \\ &= \lim_{n \to \infty} \sum_{k=1}^{n}{ \left( \frac{7-2\frac{1}{k}}{8-3 \frac{\sqrt{k}}{k}}\right)}^k \\ \end{align}$

Am I on the right track ? If so, how should I go on ? I need some help to do it without the root test.

Answer 1

HINT

Note that eventually

$$0<\frac{7k-2}{8k-3 \sqrt{k}}\le c<1$$

Answer 2

It should be \begin{align*} \lim_{k\rightarrow\infty}\left(\left(\dfrac{7k-2}{8k-3\sqrt{k}}\right)^{k}\right)^{1/k}=\dfrac{7}{8}<1, \end{align*} by root test, it is convergent.