Question

The terms of a series \(\sum_{n=1}^{\infty} a_{n}\) are defined recursively. Determine the convergence or divergence of the series. Explain your reasoning. $$ a_{1}=\frac{1}{4}, a_{n+1}=\sqrt[n]{a_{n}} $$

Short answer

The series converges by the Monotone Convergence Theorem because the terms in the series are decreasing and bounded below by 0.

Step-by-step solution

Interpret the given equations.

The first equation \( a_{1}=\frac{1}{4} \) sets the value of the first term in the series to \(\frac{1}{4}\). The second equation \( a_{n+1}=\sqrt[n]{a_{n}} \) indicates that each succeeding term is the n-th root of the preceding term.

Analyze the pattern.

Since each term \( a_{n+1} \) in the series is the n-th root of \( a_{n} \), the terms in the series will get closer and closer to 1, but they are always less than 1. Therefore, the terms in the series are decreasing.

Apply the Monotone Convergence Theorem.

Since the terms in the series are decreasing and bounded below by 0, we can use the Monotone Convergence Theorem. This theorem states that a sequence that has a lower bound and is decreasing will converge. Therefore, this series is convergent.