The problem asks for the sum of the infinite series $\sum_{n=1}^{\infty} a_n$, given that the partial sum of the series is defined by:

$s_n = 3 - 8(0.8)^n$

Solution:

To find the sum of the series, we must first understand that the total sum $S$ of the infinite series is the limit of the partial sum $s_n$ as $n \to \infty$. That is:

$S = \lim_{n \to \infty} s_n$

Step 1: Evaluate the limit of $s_n$.

$\lim_{n \to \infty} s_n = \lim_{n \to \infty} \left( 3 - 8(0.8)^n \right)$

• As $n \to \infty$, the term $(0.8)^n$ approaches 0 (because $0.8 < 1$). $\lim_{n \to \infty} \left( 3 - 8(0.8)^n \right) = 3 - 8(0) = 3$

So, the sum of the series is:

$S = 3$

Step 2: Verify the reasoning.

Since the partial sum $s_n$ converges to a finite value (3), it means the series converges, and its sum is exactly 3.

Final Answer:

$\boxed{S = 3}$

Would you like me to elaborate further on any part of this solution?

Here are 5 follow-up questions to deepen your understanding:

1. How do we know the series converges based on the behavior of $s_n$?
2. What is the relationship between partial sums and the series terms $a_n$?
3. Can you derive the general term $a_n$ from the partial sum formula?
4. How does the limit $\lim_{n \to \infty} (0.8)^n$ behave mathematically?
5. What happens if the base of the power term (e.g., 0.8) was greater than 1?

Tip: When dealing with infinite series, always check if the sequence of partial sums converges to a finite value to determine the sum.