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To determine which of the series are convergent, let's analyze each series individually using various convergence tests:

### 1. Series 1:
\[
\sum_{n=0}^{\infty} \frac{(-9)^{n+1}}{7 \cdot 2^n}
\]

This series resembles a geometric series because it has terms in the form of a constant raised to a power of \( n \). We can rewrite it as:
\[
\sum_{n=0}^{\infty} \left( -\frac{9}{2} \right)^n \frac{1}{7}
\]

Since \( \left| \frac{-9}{2} \right| > 1 \), this series diverges because the terms do not approach zero, violating the necessary condition for convergence of a geometric series.

### 2. Series 2:
\[
\sum_{n=0}^{\infty} \sin \left( \frac{4n + 1}{9n + 4} \right)
\]

For convergence, the terms of the series must approach zero as \( n \to \infty \). Examining the argument of the sine function, we have:
\[
\lim_{n \to \infty} \frac{4n + 1}{9n + 4} = \frac{4}{9}
\]
Thus, \( \sin \left( \frac{4n + 1}{9n + 4} \right) \) does not tend to zero as \( n \to \infty \), so this series **diverges**.

### 3. Series 3:
\[
\sum_{n=1}^{\infty} (-1)^n n^{-\frac{9}{4}}
\]

This is an alternating series with terms \( a_n = n^{-\frac{9}{4}} \). To test for convergence, we can apply the **Alternating Series Test** (Leibniz's Test). The requirements are:
1. \( a_n \) must be positive, decreasing, and approach zero as \( n \to \infty \).
   
Since \( n^{-\frac{9}{4}} \) is positive, decreases for \( n \geq 1 \), and \( \lim_{n \to \infty} n^{-\frac{9}{4}} = 0 \), this series converges by the Alternating Series Test.

### 4. Series 4:
\[
\sum_{n=0}^{\infty} \frac{5^{2n}}{2^{3n}}
\]

This series can be simplified as:
\[
\sum_{n=0}^{\infty} \left( \frac{5^2}{2^3} \right)^n = \sum_{n=0}^{\infty} \left( \frac{25}{8} \right)^n
\]

Since \( \frac{25}{8} > 1 \), this is a divergent geometric series because the ratio exceeds 1.

### Conclusion:
Only **Series 3** is convergent.

Would you like more details on any particular series or convergence tests?

### Related Questions
1. How does the Alternating Series Test determine convergence?
2. What other tests could we use for convergence if the Alternating Series Test fails?
3. Can all geometric series be tested for convergence simply by checking the ratio?
4. How do we determine convergence for trigonometric series like Series 2?
5. What happens to a series if its terms do not approach zero?

### Tip:
For a series to converge, especially a geometric one, ensure that the common ratio \( r \) satisfies \( |r| < 1 \).

Learn how to identify convergent series using convergence tests. Explore geometric series, alternating series, and limit-based convergence criteria for advanced college-level calculus problems.

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