<s>[INST] check if $\sum_{k=1}^{\infty}{ \left( \frac{7k-2}{8k-3 \sqrt{k}}\right)}^k$ converges 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. 
 [/INST] HINT
Note that eventually
$$0<\frac{7k-2}{8k-3 \sqrt{k}}\le c<1$$
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