Question

Determine the convergence of the given series using the Ratio Test. If the Ratio Test is inconclusive, state so and determine convergence with another test. $$\sum_{n=1}^{\infty} \frac{2 \cdot 4 \cdot 6 \cdot 8 \cdots 2 n}{3 \cdot 6 \cdot 9 \cdot 12 \cdots 3 n}$$

Short answer

The series converges by the Ratio Test as the limit is less than 1.

Step-by-step solution

Understand the Series

The given series is \( \sum_{n=1}^{\infty} \frac{2 \cdot 4 \cdot 6 \cdot 8 \cdots 2 n}{3 \cdot 6 \cdot 9 \cdot 12 \cdots 3 n} \). The numerator and denominator are products of arithmetic sequences.

Simplify the n-th Term

The n-th term \( a_n \) is \( \frac{2^n \cdot n!}{3^n \cdot (n!)^2} \). Simplifying, the numerator is obtained from an even number product, and the denominator is an arithmetic sequence with a step of 3.

Apply the Ratio Test

The Ratio Test requires evaluating \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). We calculate: \( \frac{a_{n+1}}{a_n} = \frac{2^{n+1} \cdot (n+1)!}{3^{n+1} \cdot ((n+1)!)^2} \times \frac{3^n \cdot (n!)^2}{2^n \cdot n!} \). This simplifies to \( \frac{2(n+1)}{3(n+1)} = \frac{2}{3} \).

Evaluate the Limit

The limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{2}{3} < 1 \). Since this limit exists and is less than 1, by the Ratio Test, the series converges.

Conclusion Based on the Ratio Test

The Ratio Test concludes that since the limit is less than 1, the series \( \sum_{n=1}^{\infty} a_n \) converges.

Key concepts

Convergence

Convergence is a critical concept when studying series. We say an infinite series converges if the sum of its infinite terms approaches a finite value. This means that as we continue adding more terms, the total sum settles to a specific number rather than increasing indefinitely. Different mathematical tests help us understand whether a series converges or not.

One important test is the Ratio Test. This test specifically helps determine convergence by examining the ratio of successive terms in the series. For a series represented as \( \sum a_n \), the Ratio Test inspects the limit

• \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \)

If the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. However, if the limit equals 1, the Ratio Test is inconclusive, and other tests might be required.

Understanding convergence is crucial in calculus and series analysis. It tells us whether infinite additions can produce a meaningful, finite result.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. It takes the form \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) is a term from the sequence. The infinite nature of such a series suggests that it continues endlessly, without stopping.

The study of infinite series often focuses on whether they converge or diverge. Convergence means the series has a finite sum. If a series diverges, its sum is infinite or undefined. In mathematics, the convergence of an infinite series provides valuable insights, particularly in calculus and mathematical analysis.

Different types of series exist, such as geometric series and arithmetic series, each with distinct properties and methods for determining convergence or divergence. Tests like the Ratio Test are designed to efficiently analyze the behavior of these series over infinite terms.

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference. For example, in the sequence \( 2, 4, 6, 8, \ldots \), the constant difference is 2. This sequence is formed by starting at 2 and continually adding 2.

The general form of an arithmetic sequence is \( a, a+d, a+2d, a+3d, \ldots \), where \( d \) is the common difference. In the context of the given exercise, both the numerator and denominator of the series are products derived from arithmetic sequences:

• Numerator: \( 2, 4, 6, \ldots, 2n \)
• Denominator: \( 3, 6, 9, \ldots, 3n \)

These products lead to factorial expressions when analyzing terms individually. Understanding arithmetic sequences is vital for simplifying series, such as breaking down complex fractions into manageable terms or identifying patterns in series solutions.