To find the sum of the given series:

$S = -\frac{20}{8} + \frac{80}{64} - \frac{320}{512} + \frac{1280}{4096}$

Step 1: Simplify the terms

Each term can be simplified: $-\frac{20}{8} = -2.5, \quad \frac{80}{64} = 1.25, \quad \frac{320}{512} = 0.625, \quad \frac{1280}{4096} = 0.3125$

So the series becomes: $S = -2.5 + 1.25 - 0.625 + 0.3125$

Step 2: Analyze the series

This is a geometric series with:

• First term ($a$): $-2.5$,
• Common ratio ($r$): $-\frac{1}{2}$.

Step 3: Sum of an infinite geometric series

The formula for the sum of an infinite geometric series is: $S_{\infty} = \frac{a}{1 - r}$

Here: $a = -2.5, \quad r = -\frac{1}{2}$

Substitute these values: $S_{\infty} = \frac{-2.5}{1 - (-\frac{1}{2})} = \frac{-2.5}{1 + \frac{1}{2}} = \frac{-2.5}{\frac{3}{2}} = -2.5 \cdot \frac{2}{3} = -\frac{5}{3} \approx -1.67$

Final Answer:

The sum of the infinite geometric series is: $\boxed{-\frac{5}{3} \, \text{or approximately } -1.67.}$

Would you like a breakdown of any part or further examples? 😊

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Expanding Questions:

1. How can we derive the common ratio $r$ for a geometric series?
2. What conditions must a geometric series meet to converge?
3. How does the formula for the sum of a finite geometric series differ from that of an infinite series?
4. What happens if the common ratio $|r|$ is greater than 1?
5. Can a geometric series have a mix of positive and negative terms?

Tip: For an infinite geometric series to converge, the absolute value of the common ratio must be less than 1 ($|r| < 1$).