Problem 8 - Calculus: Early Transcendentals

Question

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? \( \displaystyle \sum_{n = 1}^{\infty} \frac {(-1)^{n - 1}}{n!} \)

Step-by-Step Solution

Answer

The series appears to be convergent, with partial sums approaching approximately 0.6321.

Step 1: Understand the Sequence Terms

The given series is \( \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n!} \). For each term \( a_n \), it is defined by the formula \( a_n = \frac{(-1)^{n-1}}{n!} \). Calculate the first eight terms using this formula.

Step 2: Calculate First Eight Terms

Using the given formula, compute the first eight terms of the series:- \( a_1 = \frac{(-1)^{1-1}}{1!} = \frac{1}{1} = 1 \)- \( a_2 = \frac{(-1)^{2-1}}{2!} = \frac{-1}{2} = -0.5 \)- \( a_3 = \frac{(-1)^{3-1}}{3!} = \frac{1}{6} \approx 0.1667 \)- \( a_4 = \frac{(-1)^{4-1}}{4!} = \frac{-1}{24} \approx -0.0417 \)- \( a_5 = \frac{(-1)^{5-1}}{5!} = \frac{1}{120} \approx 0.0083 \)- \( a_6 = \frac{(-1)^{6-1}}{6!} = \frac{-1}{720} \approx -0.0014 \)- \( a_7 = \frac{(-1)^{7-1}}{7!} = \frac{1}{5040} \approx 0.0002 \)- \( a_8 = \frac{(-1)^{8-1}}{8!} = \frac{-1}{40320} \approx -0.00002 \)

Step 3: Calculate Partial Sums

Calculate the first eight partial sums \( S_n \): - \( S_1 = a_1 = 1 \)- \( S_2 = S_1 + a_2 = 1 - 0.5 = 0.5 \)- \( S_3 = S_2 + a_3 = 0.5 + 0.1667 = 0.6667 \)- \( S_4 = S_3 + a_4 = 0.6667 - 0.0417 = 0.6250 \)- \( S_5 = S_4 + a_5 = 0.6250 + 0.0083 = 0.6333 \)- \( S_6 = S_5 + a_6 = 0.6333 - 0.0014 = 0.6319 \)- \( S_7 = S_6 + a_7 = 0.6319 + 0.0002 = 0.6321 \)- \( S_8 = S_7 + a_8 = 0.6321 - 0.00002 = 0.6321 \)

Step 4: Analyze the Partial Sums

Observe the behavior of the partial sums \( S_n \) as \( n \) increases. The partial sums remain very close to a specific value (approximately 0.6321). Each new term contributes less and less to the change in the sum.

Step 5: Determine Convergence

Since the partial sums appear to approach a specific value and change very little with additional terms, the series seems to be convergent.

Key Concepts

Sequence of Partial Sums

When exploring series like \( \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n!} \), a common approach is examining the sequence of partial sums. A partial sum, denoted as \( S_n \), is the sum of the first \( n \) terms of the series. Simply put, for a series \( a_1, a_2, a_3, \ldots \), the partial sum \( S_n \) is given by:

• \( S_1 = a_1 \)
• \( S_2 = a_1 + a_2 \)
• \( S_3 = a_1 + a_2 + a_3 \)

The process continues infinitely, but examining a limited number of terms can reveal their convergence behavior. By calculating the partial sums, like in our example up to \( S_8 \), we identify if the sequence stabilizes towards a certain number or not.

Factorial Sequence

In the series we are discussing, the denominator of each term is a factorial, which plays a critical role in its behavior. A factorial, written as \( n! \), is the product of all positive integers up to \( n \). This means:

• \( 1! = 1 \)
• \( 2! = 2 \times 1 = 2 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)

As \( n \) increases, \( n! \) grows significantly larger. This rapid growth means that each successive term in the series \( \frac{(-1)^{n-1}}{n!} \) becomes quite small. This diminishing size of terms is often a clue for the convergence of a series.

Alternating Series

This particular series is an alternating series because the terms change sign regularly. This alternation arises from the \((-1)^{n-1}\) factor, making the series alternate between positive and negative values. Two key characteristics of alternating series are:

• The terms shrink in absolute size, which is due to the factorial in the denominator.
• The series can converge if the absolute value of terms \( a_n \) approaches zero as \( n \) increases.

In practical terms, the partial sums wiggle up and down, gradually fitting closer to a certain number, just as we observed with partial sums converging towards 0.6321.

Convergence Analysis

Convergence analysis involves determining whether a series approaches a specific value as the number of terms increases. For our series with partial sums \( S_1, S_2, \ldots, S_8 \), we noticed a pattern:

• The differences in consecutive partial sums became smaller, indicating the change between them lessens as \( n \) grows.
• At a certain point, the sums stopped significantly changing; this is a good sign of convergence.

To confidently conclude convergence, particularly for alternating series, we use the alternating series test conditions. Since our series terms diminish towards zero and alternate in signs, we can assert convergence, demonstrated practically by observing the cumulative sums settling around the value 0.6321.