Problem 30 - Calculus: Early Transcendentals

Question

Test the series for convergence or divergence. \( \displaystyle \sum_{j = 1}^{\infty} ( - 1)^j \frac {\sqrt{j}}{j + 5} \)

Step-by-Step Solution

Answer

The series converges by the Alternating Series Test.

Step 1: Identify the series as an alternating series

The given series is \((-1)^j \frac{\sqrt{j}}{j+5}\), which is an alternating series because of the \((-1)^j\) factor. This factor causes the terms to alternate in sign.

Step 2: Check the Alternating Series Test conditions

The Alternating Series Test states that a series \( \sum (-1)^j a_j \) converges if the sequence \( a_j \) is decreasing and \( \lim_{j \to \infty} a_j = 0 \). In this series, \( a_j = \frac{\sqrt{j}}{j+5} \).

Step 3: Test if \( a_j \) is decreasing

Examine \( a_j = \frac{\sqrt{j}}{j+5} \). To determine if it's decreasing, consider the derivative of \( f(j) = \frac{\sqrt{j}}{j+5} \). Simplify the derivative to see if \( f'(j) < 0 \) for large \( j \).

Step 4: Compute the limit \( \lim_{j \to \infty} a_j \)

Evaluate \( \lim_{j \to \infty} \frac{\sqrt{j}}{j+5} \). Notice that the numerator grows like \( \sqrt{j} \) and the denominator like \( j \). Thus, the limit is 0 as \( j \to \infty \).

Step 5: Conclusion using the Alternating Series Test

Since \( a_j = \frac{\sqrt{j}}{j+5} \) is eventually decreasing (after confirming the derivative condition) and \( \lim_{j \to \infty} a_j = 0 \), the series converges by the Alternating Series Test.

Key Concepts

Alternating Series Test

The Alternating Series Test is a fundamental tool used in analyzing the convergence of certain series. It specifically applies to series in the form of \(\sum (-1)^j a_j\), where the terms alternate in sign due to the \((-1)^j\) factor.
This test contains two main conditions for convergence:

• The sequence of absolute values \(a_j\) should be decreasing.
• The limit \(\lim_{j \to \infty} a_j\) must be 0.

The given series \((-1)^j \frac{\sqrt{j}}{j+5}\) fits the alternating series template. Applying the Alternating Series Test requires us to confirm both conditions are met for convergence.

Limit Computation

Limit computation is a key process in determining the convergence of sequences and series. In this context, we need to find the limit of the sequence \(a_j = \frac{\sqrt{j}}{j+5}\) as \(j\) approaches infinity.
To compute this limit, it's helpful to compare the growth rates of the numerator and the denominator.

• The numerator \(\sqrt{j}\) grows slower than the denominator \(j\).
• As a rule of thumb, whenever the top grows slower, the fraction goes to zero.

Therefore, \(\lim_{j \to \infty} \frac{\sqrt{j}}{j+5} = 0\), satisfying one of the conditions of the Alternating Series Test.

Derivative Test

The Derivative Test is often used to determine if a sequence is decreasing, which is one of the requirements of the Alternating Series Test. In our series, we have the function \(f(j) = \frac{\sqrt{j}}{j+5}\).
The derivative \(f'(j)\) can be calculated and simplified to assess the behavior of \(f(j)\) with respect to increase or decrease. Without diving into complex calculus, evaluating the sign of \(f'(j)\) for large \(j\) helps us see that \(f'(j) < 0\), indicating a decreasing sequence.
This fulfills the second condition of the Alternating Series Test, ensuring that \(a_j\) diminishes as \(j\) increases, leading to the eventual convergence of the series.

Sequence Behavior

Understanding sequence behavior is critical when dealing with series convergence. The term \(a_j = \frac{\sqrt{j}}{j+5}\) represents each element of our sequence.
Recognizing that the sequence is composed of rational expressions allows us to analyze its behavior as \(j\) becomes very large.

• The numerator \(\sqrt{j}\) increases slowly, while the denominator \(j+5\) increases rapidly.
• This causes \(a_j\) to head toward zero.

Both the decreasing nature confirmed by the derivative analysis and the limiting tendency \(\lim_{j \to \infty} a_j = 0\) align with required conditions for convergence, validating the method we used to analyze the series.