Answer
Both the root and ratio tests fail.
Work Step by Step
Let us consider $a_n=\dfrac{1}{(\ln n)^p}$
Apply root test.
This implies that $\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{1}{(\ln (n+1))^p}}{\dfrac{1}{(\ln n)^p}}|=\lim\limits_{n \to \infty}|\dfrac{(\ln n)^p}{[\ln (n+1)]^p}=1$
we can see that the ratio test is not satisfied, thus fails.
Let us consider $a_n=\dfrac{1}{(\ln n)^p}$
Next, need to apply root test.
This implies that $\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} (|\dfrac{1}{(\ln n)^p})^{(1/n)}=1$
Thus, we can see that the root test is not satisfied, thus fails.
