Problem 49 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the series converges absolutely, converges conditionally, or diverges. Explain your reasoning carefully. \(\sum_{k=1}^{\infty} \frac{(-1)^{k} 5^{k}}{k !}\)

Step-by-Step Solution

Answer

The given series \(\sum_{k=1}^{\infty} \frac{(-1)^{k} 5^{k}}{k !}\) converges absolutely.

Step 1: Testing for absolute convergence

To test for absolute convergence, it's essential to examine the absolute value of the series and apply the Ratio Test. The Ratio Test states if the limit of the absolute ratio between the (k+1)th term and the kth term of the series as k tends to infinity is less than one, the series is absolutely convergent. \n\nTherefore, it can be represented as: \(\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\)where \(a_k = \frac{5^{k}}{k!}\), and \(a_{k+1} = \frac{5^{k+1}}{(k+1)!}\).Plugging \(a_{k+1}\) and \(a_k\) into the Ratio Test yields: \(\lim_{k \to \infty} \left| \frac{5^{k+1}/(k+1)!}{5^{k}/k!} \right| = \lim_{k \to \infty} \frac{5}{k+1} \)As k tends to infinity, the limit goes to 0, which is less than 1. Therefore, the series is absolutely convergent.

Step 2: Mentioning conditional convergence

In case, step 1 failed i.e the series isn't absolutely convergent, the student would then need to test for conditional convergence. An absolutely convergent series is also conditionally convergent, but the reverse doesn't hold. The Alternating Series Test could help determine if the series is conditionally converging. However, since the series was found to be absolutely convergent, it's redundant to test for conditional convergence.

Key Concepts

Absolute Convergence

To understand absolute convergence, imagine checking if the absolute value of a series converges. By ignoring the signs of terms, you're focusing on the sum's magnitude.

When a series of absolute values converges, we say it's **absolutely convergent**. This is a key detail because absolute convergence implies convergence of the original series, including any alternating signs. Testing for absolute convergence often involves transforming the series into one without negative signs and using tests like the Ratio Test. For instance, in the exercise, the absolute series \(\sum_{k=1}^{\infty} \frac{5^k}{k!}\) is considered, leading us to apply the Ratio Test.

Ratio Test

The Ratio Test is a handy tool for checking absolute convergence.

It involves comparing the ratio of the absolute values of consecutive terms. If the limit of these ratios is less than 1, the series converges absolutely. In our exercise, we calculate:

• \( a_k = \frac{5^k}{k!} \)
• \( a_{k+1} = \frac{5^{k+1}}{(k+1)!} \)

The Ratio Test formula becomes: \[\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \frac{5^{k+1}/(k+1)!}{5^k/k!} = \lim_{k \to \infty} \frac{5}{k+1} \]Since this limit equals 0, which is less than 1, the series is absolutely convergent. Applying the test effectively confirms the behavior of the series across its infinite terms.

Alternating Series Test

Sometimes, a series might not be absolutely convergent, yet it still converges. That's where the Alternating Series Test comes in.

This test applies to series where the signs of terms alternate between positive and negative. These series are characterized by two primary conditions:

• Terms decrease in magnitude as the series progresses.
• The limit of the terms as they approach infinity is zero.

In the provided exercise, had the series not been absolutely convergent, these criteria would be used to test conditional convergence. However, the Ratio Test already confirmed absolute convergence, making the Alternating Series Test unnecessary in this context.

Factorials in Series

Understanding factorials is crucial when working with series, especially those involving combinations and permutations. Factorials, denoted by an exclamation mark (e.g., \(n!\)), represent the product of all positive integers up to \(n\).

For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials often appear in series related to convergence testing because they grow quite fast, affecting the limit and convergence behavior. In the given exercise, factorials in the denominator, \(k!\), stabilize the growth of the numerator \(5^k\), ensuring the fractions become smaller as \(k\) increases. This decrease is instrumental in proving convergence through the Ratio Test.