Problem 87 - Essential Calculus

Question

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $$ \sum_{n=1}^{\infty} \frac{(-3)^{n}}{3 \cdot 5 \cdot 7 \cdot \cdot(2 n+1)} $$

Step-by-Step Solution

Answer

The series \(\sum_{n=1}^{\infty} \frac{(-3)^{n}}{3 \cdot 5 \cdot 7 \cdot \cdot(2 n+1)}\) is convergent by the Ratio Test.

Step 1: Identify the Series

The series given is of form \(\frac{(-3)^{n}}{3 \cdot 5 \cdot 7 \cdot \ldots \cdot (2n+1)}\).

Step 2: Apply the Ratio Test

The Ratio Test states that for a series \(\sum a_n\), if there exists a number \(L < 1\) such that \(\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L < 1\), the series is convergent. Let's see if this holds true here: We want to calculate \(\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty} \left|\frac{(-3)^{n+1}}{3\cdot5\cdot7\cdot\ldots\cdot(2(n+1)+1)} \cdot \frac{3\cdot5\cdot7\cdot\ldots\cdot(2n+1)}{(-3)^n}\right|\). After reduction, we find that this limit is \(\lim_{n\to\infty} |-3|/((2n + 5)) = 3 / (2n+5)\). As \(n \to \infty\), this limit is equal to 0, which is less than 1.

Step 3: Draw Conclusion

Since the limit of \(|a_{n+1}/a_n|\) as \(n\) approaches infinity is less than 1, by the Ratio Test, this series is convergent.

Key Concepts

Ratio Test

The Ratio Test is a popular tool used to determine if an infinite series converges or diverges. It's applicable when you have terms that seem to shrink rapidly, either through an increasing factorial in the denominator or exponentially larger terms in the numerator.

To apply the Ratio Test to a series \( \sum a_n \), you calculate the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). If this limit \( L \) is less than 1, the series converges. If \( L > 1 \) or if the limit doesn't exist, the series diverges. If \( L = 1 \), the test is inconclusive, and you'll need to try another method.

In our given problem, we applied the Ratio Test to the series \( \sum \frac{(-3)^n}{3 \cdot 5 \cdot 7 \cdot \ldots \cdot (2n+1)} \). The calculations showed that the limit \( \lim_{n \to \infty} \frac{3}{2n+5} \) tends towards 0, a number less than 1, proving convergence.

Infinite Series

An infinite series is simply the sum of infinitely many terms. To visualize it, think about adding an endless list of numbers. Examples include the familiar geometric series or the harmonic series.

Infinite series can either converge or diverge. When a series converges, the sum approaches a certain finite value even as the number of terms becomes infinitely large. On the other hand, if a series diverges, it means the sum increases without bound or fluctuates endlessly.

For example, the series \( \sum \frac{1}{n} \) is divergent because as \( n \to \infty \), the sum grows without limit. Understanding if a series converges or diverges is crucial in many areas of calculus and mathematical analysis.

Convergence Tests

Convergence tests are methods used to determine whether an infinite series converges. Each test comes with specific criteria, making them suitable for different types of series. Some common convergence tests include:

• Ratio Test: As discussed, it works well with series whose terms include factorials or exponentials.
• Root Test: Useful for series with terms raised to the power \( n \).
• Integral Test: Applies when the series can be represented as a function that is positive, continuous, and decreasing.
• Comparison Test: Compares a series to a known benchmark series to determine convergence.
• Alternating Series Test: Specifically for series whose terms alternate in sign.

Applying these tests correctly can help solve problems like our given exercise and provide insight into the behavior of complex series. Always choose the test that aligns with the characteristics of the series you are examining.