Determine if the following series converges, diverges, or is inconclusive. 
${\sum }_{n=1}^{\infty }{\left(\frac{4n+1}{n-2}\right)}^n$

• A. Converges
• B. Diverges
• C. Inconclusive

Answer: (B)

Verified step by step guidance

1. Step 1: Recognize that the series involves a term raised to the power of n, which suggests that the root test or ratio test might be useful for determining convergence or divergence.
2. Step 2: Apply the root test. The root test involves evaluating \( ho = ext{lim}_{n \to \infty} \sqrt[n]{a_n} \), where \( a_n \) is the general term of the series. If \( ho < 1 \), the series converges; if \( ho > 1 \), the series diverges; if \( ho = 1 \), the test is inconclusive.
3. Step 3: Substitute \( a_n = \left(\frac{4n+1}{n-2}\right)^n \) into the root test formula. This simplifies to \( \rho = \text{lim}_{n \to \infty} \left(\frac{4n+1}{n-2}\right) \), since the \( n \)-th root cancels the exponent \( n \).
4. Step 4: Simplify \( \frac{4n+1}{n-2} \) as \( n \to \infty \). Divide numerator and denominator by \( n \) to get \( \frac{4 + \frac{1}{n}}{1 - \frac{2}{n}} \). As \( n \to \infty \), \( \frac{1}{n} \to 0 \) and \( \frac{2}{n} \to 0 \), so \( \frac{4 + \frac{1}{n}}{1 - \frac{2}{n}} \to 4 \).
5. Step 5: Conclude based on the root test. Since \( \rho = 4 > 1 \), the series diverges.