Problem 57 - Essential Calculus

Question

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n+1} n !}{1 \cdot 3 \cdot 5 \cdot \cdots(2 n+1)} $$

Step-by-Step Solution

Answer

The series is convergent.

Step 1: Define the series and the Ratio Test

The series in the task is \( \sum_{n=0}^{\infty} \frac{(-1)^{n+1} n !}{1 \cdot 3 \cdot 5 \cdot \cdots (2n+1)} \). The Ratio Test states that for a series \( \sum_{n=1}^{\infty} u_n \), it is convergent if \( \lim_{n \to \infty} \left|\dfrac{u_{n+1}}{u_n}\right| < 1 \), and divergent if the limit exceeds 1.

Step 2: Calculate the ratio of consecutive terms

The ratio of the terms \( (n+1) \) and \( n \) is: \[\left|\frac{u_{n+1}}{u_n}\right| = \left|\frac{(-1)^{(n+1)+1} (n+1) !}{1 \cdot 3 \cdot 5 \cdot \cdots (2(n+1)+1)} \cdot \frac{1 \cdot 3 \cdot 5 \cdot \cdots (2n+1)}{(-1)^{n+1} n!}\right|= \left|\frac{(-1)(n+1)n!}{2(n+1)+1}\right|\]= \(\left|\frac{-(n+1)}{2n + 3}\right|\)

Step 3: Determine the limit as n approaches infinity

The value of the limit as \( n \rightarrow \infty \) is:\[\lim_{n \rightarrow \infty} \left|\frac{-(n+1)}{2n + 3}\right| = \lim_{n \rightarrow \infty} \left|- \frac{1}{2} \right| = \frac{1}{2}\]The absolute value of the limit is less than 1.

Step 4: Conclude the test

Since the absolute value of the limit is less than 1, according to the Ratio Test, the given series is convergent.

Key Concepts

Convergence of Series

Understanding the convergence of a series involves determining if the sum of an infinite series approaches a finite value as more terms are added. In many mathematical and practical applications, identifying convergence is crucial.

The Ratio Test is one of the most straightforward methods to assess convergence. This test applies to a series \( \sum_{n=1}^{\infty} u_n \) by examining the limit \( \lim_{n \to \infty} \left| \dfrac{u_{n+1}}{u_n} \right| \). Here are three key outcomes:

• If the limit is less than 1, the series converges.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test fails to indicate anything conclusive.

In our exercise, calculating the limit \( \lim_{n \to \infty} \left| \frac{- (n+1)}{2n+3} \right| \) gives us \( \frac{1}{2} \), indicating convergence.

The notion of convergence not only tells us if a series settles to a number but informs us about potential calculations or predictions based on this series.

Factorial Notation

Factorial notation, represented by the exclamation mark \(!\), is a mathematical shorthand for multiplying a series of descending natural numbers. For example, \(n!\) equals \(n \times (n-1) \times (n-2) \times \ldots \times 1\). Factorials grow rapidly as \(n\) increases, and they often appear in contexts involving permutations or combinations.

In the discussed series, each term includes the factorial \(n!\). Understanding how factorials change between terms is crucial for applying the Ratio Test. In this exercise, when analyzing \(\frac{u_{n+1}}{u_n}\), the \((n+1)!\) in the numerator comes from incrementing the factorial by one. Additionally, the drastic growth of factorials often influences the convergence behavior of series.

Factorial notation simplifies calculations involving sequential products and plays a significant role in combinatorial mathematics and convergence testing.

Alternating Series

An alternating series is one where successive terms change signs, for example, from positive to negative. Mathematically, an alternating factor like \((-1)^{n+1}\) creates these changes in sign. Alternating series are intriguing because they can converge under certain conditions, even if the non-alternating counterpart does not.

In the provided series, we have \((-1)^{n+1}\) causing the alternating nature by producing minus signs for odd indices. Such characteristics are vital while applying tests like the Ratio Test, as they affect the absolute values being tested, and a special Alternating Series Test can also be used in some cases.

Understanding alternating series is essential because the convergence of these series brings up interesting properties that could be applied to practical problem-solving scenarios, particularly in physics and engineering where oscillations and waves are modeled.