Answer
Both the root and ratio tests fail.
Work Step by Step
Consider $a_n=\dfrac{1}{(\ln n)^p}$
Now, $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{1}{(\ln (n+1))^p}}{\dfrac{1}{(\ln n)^p}}|=\lim\limits_{n \to \infty}|\dfrac{(\ln n)^p}{(\ln (n+1))^p}=1$
Thus, the ratio test does not satisfy.
Now, apply the root test.
By the Root Test $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
$l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} (|\dfrac{1}{(\ln n)^p})^{1/n}=1$
Thus, the root test also fails.