University Calculus: Early Transcendentals (3rd Edition)

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