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To find the sum of the infinite geometric series \( 5 - \frac{5}{7} + \frac{5}{49} - \frac{5}{343} + \dots \), let's break it down step by step:

### Formula for the Sum of an Infinite Geometric Series:
\[
S = \frac{a}{1 - r}
\]
where:
- \(a\) is the first term of the series.
- \(r\) is the common ratio.

### Step 1: Identify \(a\) and \(r\)
From the series:
\[
a = 5 \quad \text{(the first term of the series)}
\]

To find the common ratio \(r\):
\[
r = \frac{\text{second term}}{\text{first term}} = \frac{-\frac{5}{7}}{5} = -\frac{1}{7}.
\]

Thus, the common ratio is \(r = -\frac{1}{7}\).

### Step 2: Check if the Series Converges
For the series to converge, \(|r| < 1\). Here:
\[
|r| = \left|-\frac{1}{7}\right| = \frac{1}{7} < 1.
\]
The series converges.

### Step 3: Plug Values into the Formula
\[
S = \frac{a}{1 - r} = \frac{5}{1 - \left(-\frac{1}{7}\right)} = \frac{5}{1 + \frac{1}{7}}.
\]

Simplify the denominator:
\[
1 + \frac{1}{7} = \frac{7}{7} + \frac{1}{7} = \frac{8}{7}.
\]

Thus:
\[
S = \frac{5}{\frac{8}{7}} = 5 \cdot \frac{7}{8} = \frac{35}{8}.
\]

### Final Answer:
\[
\boxed{\frac{35}{8}}
\]

Would you like a more detailed explanation or further assistance? Here are 5 related questions to explore:

1. What happens to the sum if the common ratio \(r\) is greater than 1?
2. How do you determine whether a geometric series converges or diverges?
3. What is the formula for the sum of a finite geometric series?
4. Can you generalize the formula for an alternating geometric series?
5. What is the significance of the common ratio in geometric series?

**Tip**: Always ensure \(|r| < 1\) before applying the sum formula for infinite geometric series!

Find the sum of the infinite geometric series: 5 - 5/7 + 5/49 - 5/343 + ...

Discover how to find the sum of the infinite geometric series 5 - 5/7 + 5/49 - 5/343 + ... with a step-by-step solution. Learn about convergence, the sum formula, and its application. Perfect for Grades 10-12 students.

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