https://r2.erweima.ai/mathbot/image/1726196545733oyiuyvg7.jpg

The image contains two infinite geometric series problems. Let's solve them one by one.

### First Series:
\[
\left( \frac{-1}{5} \right) + \left( \frac{-1}{5} \right)^2 + \left( \frac{-1}{5} \right)^3 + \cdots
\]
This is an infinite geometric series with the first term \(a = \frac{-1}{5}\) and common ratio \(r = \frac{-1}{5}\).

The sum of an infinite geometric series is given by the formula:
\[
S = \frac{a}{1 - r}, \quad \text{where} \ |r| < 1
\]
Here, \(a = \frac{-1}{5}\) and \(r = \frac{-1}{5}\), so the sum becomes:
\[
S = \frac{\frac{-1}{5}}{1 - \frac{-1}{5}} = \frac{\frac{-1}{5}}{1 + \frac{1}{5}} = \frac{\frac{-1}{5}}{\frac{6}{5}} = \frac{-1}{6}
\]
Thus, the sum of the first series is:
\[
\boxed{-\frac{1}{6}}
\]

### Second Series:
\[
\sum_{i=1}^{\infty} \left( \frac{-4}{9} \right)^i
\]
This is another infinite geometric series with the first term \(a = \frac{-4}{9}\) and common ratio \(r = \frac{-4}{9}\).

Applying the sum formula for an infinite geometric series:
\[
S = \frac{a}{1 - r}, \quad \text{where} \ |r| < 1
\]
Here, \(a = \frac{-4}{9}\) and \(r = \frac{-4}{9}\), so the sum becomes:
\[
S = \frac{\frac{-4}{9}}{1 - \frac{-4}{9}} = \frac{\frac{-4}{9}}{\frac{13}{9}} = \frac{-4}{13}
\]
Thus, the sum of the second series is:
\[
\boxed{\frac{-4}{13}}
\]

### Final Answers:
1. \(-\frac{1}{6}\) for the first series.
2. \(-\frac{4}{13}\) for the second series.

Would you like a more detailed explanation or further clarifications?

Here are 5 related questions:
1. How do you determine if a series converges?
2. What are the conditions for applying the infinite geometric series formula?
3. Can you explain how the common ratio affects convergence?
4. What happens if the common ratio is greater than or equal to 1?
5. How can the formula for finite geometric series differ from the infinite case?

**Tip**: Always check that the absolute value of the common ratio is less than 1 for an infinite geometric series to converge.

Calculate the sum of the infinite geometric series: ( (-1/5) + (-1/5)^2 + (-1/5)^3 + ...) and the sum of the series: sum from i=1 to infinity of ((-4/9)^i).

Learn how to calculate the sums of infinite geometric series with negative common ratios. This detailed solution explains the convergence criteria and uses the infinite geometric series formula to find the sums: -1/6 and -4/13. Suitable for high school students in Grades 9-12.

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