Problem 21 - Calculus: Early Transcendentals

Question

Use the Ratio Test to determine whether the series is convergent or divergent. \( 1 - \frac {2!}{1 \cdot 3} + \frac {3!}{1 \cdot 3 \cdot 5} - \frac {4!}{1 \cdot 3 \cdot 5 \cdot 7} + \cdot \cdot \cdot \) \( ( - 1 )^{n-1} \frac {n!}{1 \cdot 3 \cdot 5 \cdot \space \cdot \cdot \cdot \space \cdot (2n - 1)} + \cdot \cdot \cdot \)

Step-by-Step Solution

Answer

The series converges.

Step 1: Identify the General Term

The general term of the series is given by \( a_n = (-1)^{n-1} \frac{n!}{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1)} \). This term has an alternating sign due to \((-1)^{n-1}\).

Step 2: Apply the Ratio Test Formula

The Ratio Test requires us to find \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). Start by identifying \( a_{n+1} = (-1)^n \frac{(n+1)!}{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n+1)} \).

Step 3: Compute the Ratio \( \frac{a_{n+1}}{a_n} \)

The expression for the ratio is \( \frac{(-1)^n \frac{(n+1)!}{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n+1)}}{(-1)^{n-1} \frac{n!}{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1)}} = \frac{(n+1)!}{n!} \cdot \frac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1)}{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n+1)} \cdot (-1).\) This simplifies to \((n+1) \cdot \frac{1}{(2n+1)} \cdot (-1).\)

Step 4: Simplify and Take the Limit

Simplify the expression to \((-1)\cdot \frac{n+1}{2n+1}\). The limit is \( \lim_{n \to \infty} \left| (-1)\cdot \frac{n+1}{2n+1} \right| = \lim_{n \to \infty} \frac{n+1}{2n+1} = \frac{1}{2}.\)

Step 5: Analyze the Ratio Test Result

Since the limit \( \frac{1}{2} \) is less than 1, the Ratio Test indicates that the series converges.

Key Concepts

Series Convergence

Understanding whether a series converges or diverges is an essential part of calculus. Series convergence means that as you add more terms, the sum of the series approaches a specific number, or limit. It's like zooming in on a landscape; as you get closer, the view becomes clearer until you reach a well-defined conclusion.
In our exercise, we applied the Ratio Test to the series. The Ratio Test is a powerful tool to determine convergence, primarily looking at the absolute value of the ratio of consecutive terms. If this ratio is less than one as the number of terms goes to infinity, the series converges.

• The Ratio Test provides a clear yes or no about convergence.
• There are other convergence tests, but this one is well-suited for series with factorials and alternating terms.

Knowing that a series converges is crucial, as it informs us about the behavior of potentially infinite sums, making predictions and analysis more manageable.

Alternating Series

Alternating series have terms changing sign regularly, often flipping between positive and negative. Such series introduce an interesting twist into the study of convergence.
Our given series is an alternating series, evident from the \((-1)^{n-1}\) factor in the general term. This conveys that each term alternates in sign, which is important because alternating series can potentially converge even when their non-alternating counterparts do not.
Alternating series turn formulas and approaches in unique directions:

• The Alternating Series Test is a method to confirm convergence, but here, we used the Ratio Test because of the complexity and factorials involved.
• Alternating signs can act as a balancing act, bringing the series closer to a specific value.

If the absolute value of these terms gets smaller and smaller, such as approaching zero, the series is likely to take a step toward convergence.

Factorials in Series

Factorials, denoted as \(n!\), are multiplications of all natural numbers up to \(n\). They increase very quickly and are often integral to complex series like the one in our exercise. Factorials contribute to the complexity of convergence analysis.

Here, the series includes factorials in both numerator and denominator, complicating the Ratio Test. In practice, understanding how factorials work can clarify the behavior of a series:

• Factorials grow extremely rapidly, impacting the size and rate of change of series terms.
• When combined in ratios or products, factorials can indicate how terms balance each other.

Typically, larger numerators (due to factorials) can be mitigated by similarly large factors in denominators, helping guide automatically toward convergence.

Limits in Calculus

Limits are fundamental to understanding calculus, functioning as a tool to assess behavior as variables approach certain points. They are critical in determining the outcome of convergence tests such as the Ratio Test.

In the step by step analysis, the limit \(\lim_{n \to \infty} \left| \frac{n+1}{2n+1} \right|\) was calculated, arriving at \frac{1}{2}\. This result revealed valuable insights:

• Knowing the limit allows us to infer about the behavior of a series as it proceeds to infinity.
• Essentially, if a series' term ratio approaches a number less than one, it points to convergence as seen here.

Limits illuminate the tendency of terms, showing whether they diminish or continue growing indefinitely, offering a pathway to definitively understand series behavior.