Problem 15 - Essential Calculus

Question

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n !}{1 \cdot 3 \cdot 5 \cdot \cdot \cdot(2 n-1)} $$

Step-by-Step Solution

Answer

The given series converges.

Step 1: Apply the Ratio Test

The Ratio Test requires to find the limit as \( n \) approaches infinity of the absolute value of the ratio of the \( n+1 \) term to the \( n \) term of the series.\n Thus, apply the ratio test which states:\[\lim_{{n \to \infty}} \frac{a_{n+1}}{a_n} = L\]The terms of the series \( a_n \) and \( a_{n+1} \) are:\[a_n= \frac{(-1)^{n+1} n !}{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2 n-1)}\]\[a_{n+1}= \frac{(-1)^{n+2} (n+1) !}{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n+1)}\]

Step 2: Calculate the limit

Divide \( a_{n+1} \) by \( a_n \) and apply limit as \( n \) approaches infinity.\[L= \lim_{n \to \infty} \left| \frac{(-1)^{n+2} (n+1) ! / 1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2 n+1)}{(-1)^{n+1} n ! / 1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2 n-1)}\right|\]Simplify\[L= \lim_{n \to \infty} \left| \frac{(-1)(n+1)(2n-1)!}{(2n+1)!}\right|\]\[L= \lim_{n \to \infty} \left| \frac{-1}{2n+1}\right|\]

Step 3: Interpret the result

Since the absolute value of \( L \) is zero (which is less than 1), the given series converges by the Ratio Test. Hence, it results in an alternating series, each of whose terms is smaller than the one before.

Key Concepts

Ratio Test

The Ratio Test is a method in calculus used to determine whether a series converges or diverges. This test involves calculating the limit of the absolute value of the ratio of consecutive terms in the series as the number of terms goes to infinity.
The formal statement of the Ratio Test is:

• Given a series \(\sum_{n=1}^{\infty} a_n \), compute the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).
• If \(L < 1\), the series converges.
• If \(L > 1\), or if \(L = \infty\), the series diverges.
• If \(L = 1\), the test is inconclusive.

In the given series, applying the Ratio Test involves calculating the limit of the ratio of the \(n+1\) term to the \(n\) term. If the value of \(L\) is zero, which means the ratio of terms becomes infinitely small, indicating the series converges. This is what we observe in the original step-by-step solution where \(L\) resulted in zero.

Alternating Series

An alternating series is a series where the terms alternate in sign. This means positive and negative terms appear sequentially. The series in the original exercise is an example of an alternating series because of the factor \( (-1)^{n+1} \), which changes the sign of each successive term.
To determine the convergence of such a series, besides the Ratio Test, we can also use the Alternating Series Test. This test states that:

• For the series \(\sum (-1)^{n+1} b_n\), if the absolute value of terms \(b_n\) decrease steadily, i.e., \(b_{n+1} \leq b_n\), and \(\lim_{n \to \infty} b_n = 0\), then the series converges.

In our example, each term is smaller in magnitude than the previous one, and the term's limit goes to zero, confirming convergence. Alternating series often converge even if conditions are not strictly tightened like other tests, owing to the canceling effect of terms.

Factorial Expression

Factorial expressions may seem daunting, but they are simply products of an integer and all the integers before it.
For instance, the factorial notation \(n!\) means \(n \times (n-1) \times (n-2) \times ... \times 1 \). In the context of series, factorials often appear in sequences because they grow very fast, which affects the convergence of a series.
In the given example, the presence of factorials like \(n!\) and the product of odd numbers, such as \(1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1)\), creates a complex balance.

• The denominator grows quickly, which makes each term smaller as \(n\) increases, assisting convergence.
• The numerator, which involves \(n!\), also grows but not as fast due to the odd factorial pattern on the denominator.

Understanding factorial expression is critical in recognizing growth rates that influence whether a series converges or diverges.