Answer
(a) $p = \dfrac{4}{3}$; the series converges.
(b) $p = \dfrac{1}{4}$; the series diverges.
(c) $p = \dfrac{5}{3}$; the series converges.
(d) $p = \pi $; the series converges.
Work Step by Step
(a) $\mathop \sum \limits_{k = 1}^\infty {k^{ - 4/3}} = \mathop \sum \limits_{k = 1}^\infty \dfrac{1}{{{k^{4/3}}}}$
$p = \dfrac{4}{3}$
By Theorem 9.4.5, the series converges.
(b) $\mathop \sum \limits_{k = 1}^\infty \dfrac{1}{{\sqrt[4]{k}}} = \mathop \sum \limits_{k = 1}^\infty \dfrac{1}{{{k^{1/4}}}}$
$p = \dfrac{1}{4}$
By Theorem 9.4.5, the series diverges.
(c) $\mathop \sum \limits_{k = 1}^\infty \dfrac{1}{{\sqrt[3]{{{k^5}}}}} = \mathop \sum \limits_{k = 1}^\infty \dfrac{1}{{{k^{5/3}}}}$
$p = \dfrac{5}{3}$
By Theorem 9.4.5, the series converges.
(d) $\mathop \sum \limits_{k = 1}^\infty \dfrac{1}{{{k^\pi }}}$
$p = \pi $
By Theorem 9.4.5, the series converges.
