Final Answer:
The given series 1 + 1/(8√2) + 1/(27√3) + 1/(64√4) + 1/(125√5) + … is a p-series with the general term where p = 3/2. According to Theorem 9.11, the p-series converges if p > 1 and diverges if p ≤ 1. In this case, p = 3/2 > 1, so the series converges.
Step-by-step explanation:
The given series can be expressed as Σ from n = 1 to ∞. To determine its convergence or divergence, we can apply Theorem 9.11, which states that the p-series converges if p > 1 and diverges if p ≤ 1. In the given series, the exponent p is 3/2. Since 3/2 > 1, the conditions of Theorem 9.11 for convergence are met. Therefore, the series converges.
Understanding the convergence of p-series is essential in calculus. The convergence of this particular series is determined by the exponent 3/2, indicating that the sum of the series converges to a finite value. This convergence result aligns with the conditions set by Theorem 9.11, providing a mathematical basis for analyzing the convergence behavior of p-series.