Root Test for Convergence or Divergence (ln problem)

Question

I'm stuck at this problem. How would I approach this (I have to use the root test for this one)? The ln and e and my power is throwing me off.

This is how far I have gotten.

Answer 1

I will just use the test for divergence. Take the limit

$$ \displaystyle\lim_{n \to +\infty} \left [ \ln \left (e^9 + \frac{1}{n} \right) \right ]^{n+8}$$. That clearly does not go to zero. Hence the series is divergence.

(You can't use L'Hopital's here because it is not an indeterminate form.)

EDIT: I didn't notice that the solution has to be by root test.

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By ROOT TEST:

$$ \displaystyle\lim_{n \to +\infty} \left [ \ln \left (e^9 + \frac{1}{n} \right) \right ]^{\dfrac{n+8}{n}}$$

So the exponent should read: $1+ \dfrac{8}{n}$. So as $n \to \infty$, the result is clearly greater than 1. Hence, the series is divergence.