Let's analyze the given infinite series and express it using sigma notation.

The series is: $9 + \frac{9}{4} + \frac{9}{16} + \frac{9}{64} + \frac{9}{256} + \cdots$

This is a geometric series where each term decreases by a factor of $\frac{1}{4}$. We can identify:

• The first term $a = 9$.
• The common ratio $r = \frac{1}{4}$.

Each term in the series can be written as: $9 \times \left( \frac{1}{4} \right)^n$ where $n = 0$ for the first term.

Sigma notation

The sigma notation for the given series is: $\sum_{n=0}^{\infty} 9 \left( \frac{1}{4} \right)^n$

Thus, the infinite series can be written as: $\sum_{n=0}^{\infty} 9 \left( \frac{1}{4} \right)^n$

Let me know if you'd like a further explanation, or if this clears things up! Here are five related questions to deepen your understanding:

1. What is the sum of the infinite geometric series $\sum_{n=0}^{\infty} ar^n$ when $|r| < 1$?
2. How would the sigma notation change if the series started at $n=1$?
3. What happens to the sum of the series if the common ratio $r$ is greater than 1?
4. How do you determine if a geometric series converges or diverges?
5. What is the relationship between geometric series and exponential growth/decay?

Tip: For a geometric series to converge, the absolute value of the common ratio $|r|$ must be less than 1.