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The Ratio Test is a valuable tool in determining the radius of convergence for a series. It helps us see how the terms in a series behave as they approach infinity. To use the Ratio Test, consider the series \( \sum a_n \), where \( a_n \) is a general term of the series. The test involves calculating the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). If this limit is less than 1, the series absolutely converges. If it's greater than 1, the series diverges. When the limit equals 1, the test is inconclusive.In the context of this exercise, the general term \( a_n \) is a function of factorial expressions, which makes simplification crucial. The Ratio Test requires us to derive \( a_{n+1} \) from \( a_n \) and then simplify the expression \( \frac{a_{n+1}}{a_n} \) before taking the limit. Doing so helps us find the value of \( x \) for which the series converges. This value, \( R \), forms the boundary for convergence and divergence, known as the radius of convergence.

An infinite series is a sum of an infinite sequence of terms. In mathematical notation, an infinite series is written as \( \sum_{n=1}^{\infty} a_n \), where each \( a_n \) represents the terms to be summed.Such series can behave in varied ways, sometimes converging to a finite value and sometimes diverging. Convergence means adding more terms leads closer to a specific value, whereas divergence means the sum grows without bound as more terms are added.An important aspect of handling infinite series is determining their convergence or divergence. For this task, techniques like the Ratio Test are applied to analyze the infinite behavior of the series, ensuring that \( \sum a_n \) converges within certain boundaries determined by \( |x| < R \), where \( R \) is the radius of convergence.This mathematical analysis finds practical applications across multiple scientific disciplines, proving crucial in fields ranging from engineering to physics, where modelling continuous systems or processes involves infinite sums.

Factorial sequences are often seen in series involving terms with factorial expressions, like \( n! \) or products of sequences similar to factorials, such as double factorials \( (2n)!! \). In our exercise, the sequence features both, forming a complex term.Factorials grow very rapidly compared to regular products, as they multiply all integer values up to a number \( n \). Understanding their behavior helps simplify expressions in series and is essential for applying tests like the Ratio Test. Factorial sequences are written using exclamation marks for ease: \( n! = n \times (n-1) \times \ldots \times 1 \).In more complex sequences, pieces like double factorials account for even or odd number products alone, such as \( (2n)!! = 2 \times 4 \times 6 \times \cdots \times 2n \). Knowing how to simplify these using basic properties like \( (2n)! \approx \sqrt{2\pi n} \left( \frac{n}{e} \right)^n \) via Stirling’s approximation is vital in evaluating the behavior of sequences and determining convergence through the Ratio Test.