Convergence of the series:

To determine whether the series √(1/4) √(2/6) √(3/8) ⋯ is convergent or divergent, we need to analyze the behavior of the terms of the series.

Identifying the pattern:

Looking at the terms of the series, we can observe a pattern. Each term in the series can be written as √(n/(4n-2)), where n represents the position of the term in the series.

Applying the limit test:

To check for convergence, we can apply the limit test. If the limit of the terms of the series as n approaches infinity is finite and nonzero, then the series converges. On the other hand, if the limit is zero or infinite, the series diverges.

Calculating the limit:

Let's calculate the limit of the terms of the series as n approaches infinity:

lim(n→∞) √(n/(4n-2))

We can simplify this expression by rationalizing the denominator:

lim(n→∞) √(n/(4n-2)) * √((4n-2)/(4n-2))

Simplifying further:

lim(n→∞) √((n(4n-2))/(16n^2-8n-8n+4))

lim(n→∞) √((4n^2-2n)/(16n^2-16n+4))

Taking the square root:

lim(n→∞) ((2n√(1-1/n))/(4n√(1-1/n)-2))

As n approaches infinity, the term 1/n approaches zero, so we can simplify the expression further:

lim(n→∞) ((2n√(1-0))/(4n√(1-0)-2))

lim(n→∞) ((2n)/(4n-2))

Now, we can calculate the limit:

lim(n→∞) ((2n)/(4n-2)) = 1/2

Convergence:

Since the limit of the terms of the series is finite and nonzero (1/2), the series is convergent.

Conclusion:

The series √(1/4) √(2/6) √(3/8) ⋯ is convergent.