To analyze the convergence or divergence of this series, let's first rewrite it in a general form.

The series presented is: $\frac{8}{9} - \frac{8}{11} + \frac{8}{13} - \frac{8}{15} + \frac{8}{17} - \cdots$

From Step 1, we can observe that this is an alternating series, and it suggests a summation form: $\sum_{n=1}^{\infty} (-1)^{n-1} \frac{8}{2n+7}$

Step-by-Step Analysis:

1. Identify the Terms: The general term for the series is given as $\frac{8}{2n+7}$.

2. Alternating Series Test: To determine convergence, we can apply the Alternating Series Test (Leibniz's Test), which states that if a series has terms of the form $(-1)^{n} a_n$ and:

   • $a_n$ is positive,
   • $a_n$ is decreasing,
   • $\lim_{n \to \infty} a_n = 0$, then the series converges.

3. Apply the Conditions:

   • Positivity: The terms $\frac{8}{2n+7}$ are positive.
   • Decreasing: As $n$ increases, $2n+7$ grows, so $\frac{8}{2n+7}$ decreases.
   • Limit to Zero: $\lim_{n \to \infty} \frac{8}{2n+7} = 0$.

Since all three conditions are satisfied, the series converges by the Alternating Series Test.

Would you like further details or explanations on any specific part?

Here are some related questions for practice:

1. How do you apply the Alternating Series Test for convergence?
2. What are other tests for convergence, such as the Ratio Test or Root Test?
3. How do you determine if a non-alternating series converges or diverges?
4. What is the difference between absolute and conditional convergence?
5. Can a series diverge even if its terms approach zero?

Tip: For alternating series, always verify if the terms decrease monotonically and approach zero to apply the Alternating Series Test effectively.