Final answer:
The convergent case of the harmonic p series (1/(n^p)) depends on the value of p. The series converges when p > 1 or p ≤ 1, and diverges when p = 1 or p < 1.
Step-by-step explanation:
The convergent case of the harmonic p series (1/(n^p)) can be defined as follows:
A) The series converges when p > 1.
When p is greater than 1, the terms of the series approach zero as n approaches infinity, which leads to convergence. For example, the series 1/n^2 converges to a finite value as n gets larger.
B) The series converges when p ≤ 1.
When p is less or equal to 1, the terms of the series do not approach zero as n approaches infinity, resulting in divergence. For example, the series 1/n does not converge since its terms do not tend to zero.
C) The series converges when p = 1.
When p is equal to 1, the series is known as the harmonic series, and it diverges. The terms of the series do not approach zero, causing the series to diverge.
D) The series converges when p < 1.
When p is less than 1, the terms of the series do not approach zero as n approaches infinity, resulting in divergence. For example, the series 1/n^0.5 diverges since its terms do not tend to zero.