Final answer:
To determine the convergence of the series 1n^p, we can use the limit comparison test. The series converges when p > 1 and diverges when p < 1, p ≤ 1, or p = 1.
Step-by-step explanation:
To find the values of p for which the series sum_n=1^[infinity] 1n^p is convergent, we need to consider the convergence of the series. For a series to converge, the terms of the series must approach zero as n approaches infinity. In this case, the terms of the series are 1n^p. We can use the limit comparison test to determine the convergence or divergence of the series.
Case 1: p > 1
When p > 1, the limit as n approaches infinity of 1n^p is zero. Therefore, the series converges.
Case 2: p < 1
When p < 1, the limit as n approaches infinity of 1n^p is infinity. Therefore, the series diverges.
Case 3: p ≤ 1
When p ≤ 1, the limit as n approaches infinity of 1n^p is infinity. Therefore, the series diverges.
Case 4: p = 1
When p = 1, the limit as n approaches infinity of 1n^p is infinity. Therefore, the series diverges.