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Justify that the three conditions of the alternating series test for the following infinite series is true.

To justify that the three conditions of the alternating series test are true for the given infinite series:

\[
\sum_{j=1}^{\infty} \frac{(-1)^j}{\sqrt{j + 17}},
\]

we must verify the following three conditions:

### 1. **Alternating Sign Condition:**
   - The series must alternate in sign, which is ensured by the factor \((-1)^j\).
   - In this series, the term \(\frac{(-1)^j}{\sqrt{j + 17}}\) does indeed alternate in sign for each consecutive value of \(j\) since \((-1)^j\) alternates between \(-1\) and \(1\). 
   - **Conclusion:** The alternating sign condition is satisfied.

### 2. **Monotonic Decreasing Condition:**
   - The absolute value of the terms \(\frac{1}{\sqrt{j + 17}}\) must be monotonically decreasing.
   - Consider the function \(a_j = \frac{1}{\sqrt{j + 17}}\). To show that \(a_j\) is decreasing, we check if \(a_j > a_{j+1}\) for all \(j\).
   - Specifically:
     \[
     a_j = \frac{1}{\sqrt{j + 17}}, \quad a_{j+1} = \frac{1}{\sqrt{(j+1) + 17}} = \frac{1}{\sqrt{j + 18}}.
     \]
     - We need to verify that \(a_j > a_{j+1}\), which is equivalent to verifying that:
     \[
     \frac{1}{\sqrt{j + 17}} > \frac{1}{\sqrt{j + 18}}.
     \]
     - Since \(\sqrt{j + 17} < \sqrt{j + 18}\), it follows that \(\frac{1}{\sqrt{j + 17}} > \frac{1}{\sqrt{j + 18}}\).
   - **Conclusion:** The terms are monotonically decreasing.

### 3. **Limit Condition:**
   - The limit of the absolute value of the terms \(a_j\) as \(j \to \infty\) must be zero.
   - We need to compute:
     \[
     \lim_{j \to \infty} \frac{1}{\sqrt{j + 17}}.
     \]
     - As \(j \to \infty\), \(j + 17 \to \infty\), so:
     \[
     \lim_{j \to \infty} \frac{1}{\sqrt{j + 17}} = 0.
     \]
   - **Conclusion:** The limit condition is satisfied.

### **Final Conclusion:**
Since all three conditions of the alternating series test are satisfied (alternating sign, monotonic decrease, and limit to zero), the series:

\[
\sum_{j=1}^{\infty} \frac{(-1)^j}{\sqrt{j + 17}}
\]

converges by the Alternating Series Test.

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Would you like more details or have any questions? Here are five related questions:

1. How would the series change if the exponent in \((-1)^j\) were different?
2. What would happen if the denominator were not a square root function?
3. How can the Alternating Series Remainder Theorem be applied to approximate the sum of this series?
4. What is the significance of each condition in the Alternating Series Test?
5. How does the rate of decrease of the sequence affect the convergence?

**Tip:** When applying the Alternating Series Test, always ensure that each condition is independently verified to avoid mistakes in determining convergence.

Learn how to justify the three conditions of the Alternating Series Test for the series sum((-1)^j / sqrt(j + 17)) using mathematical analysis. This detailed explanation covers the alternating sign condition, monotonic decreasing condition, and limit condition, ensuring thorough understanding for advanced high school students.

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